Method and system for moving boundary unsteady flow field prediction based on cross-time map structure

Abstract

The application discloses a moving boundary unsteady flow field prediction method and system based on a cross-time graph structure, defines nodes corresponding to node data of a source time as a source node set, and defines nodes corresponding to node positions of a target time as a target node set; a cross-time bipartite graph is constructed based on the source node set and the target node set; source node simultaneous time space edges are constructed within the source node set, and target node simultaneous time space edges are constructed within the target node set; geometric identity features of each target node in the target node set are constructed; node coordinates of the source node set, node physical quantities of the source node set, node coordinates of the target node set, geometric identity features of the target node set, the cross-time bipartite graph, the source node simultaneous time space edges and the target node simultaneous time space edges are input into a graph neural network model that is obtained through pre-training, and predicted flow field physical quantities of the target node set are obtained, so that the prediction accuracy of a dynamic mesh flow field is improved.

Description

Technical Field

[0001] This application relates to the fields of numerical simulation of fluid mechanics, dynamic boundary flow field modeling, deep learning order reduction model and graph neural network cross technology, and in particular to a method and system for predicting unsteady dynamic boundary flow fields based on cross-time graph structure. Background Technology

[0002] Flexible dynamic boundary structures such as flexible biomimetic underwater vehicles, wave-shaped airfoils, and manta ray-inspired propulsion devices induce unsteady flow fields during operation due to continuous boundary deformation, which includes complex flow phenomena such as boundary layer evolution, vortex shedding, and wake development. To obtain high-precision flow field information, computational fluid dynamics (CFD) methods are typically used for high-fidelity numerical simulations. However, high-fidelity CFD simulations are computationally expensive and involve large datasets, making them difficult to directly apply to scenarios such as parameter optimization, real-time control, or closed-loop design. Therefore, there is an urgent need to construct reduced-order models capable of rapidly predicting unsteady flow fields.

[0003] Existing convolutional neural network-based order reduction models typically assume that flow field data resides on a fixed, regular grid. Under conditions of dynamic grid deformation, reconstruction, resampling, or fixed spatial window truncation, the node indices of adjacent time-series node sets cannot guarantee corresponding to the same spatial location or functional region. Therefore, a stable cross-time-series information transmission relationship cannot be directly established based on node indices. Under conditions of continuous deformation of flexible boundaries, node positions change constantly with the dynamic boundary and grid reconstruction, and sampling points within a fixed spatial window are also locally replaced over time. Directly assuming a stable correspondence between nodes with the same index at adjacent time points easily leads to errors in physical quantity propagation, resulting in a significant decrease in prediction accuracy near the boundary and in the wake region, making it difficult to meet the demand for high-precision and rapid prediction of complex dynamic boundary flow fields. Summary of the Invention

[0004] The main objective of this disclosure is to propose a method and system for predicting unsteady flow fields at moving boundaries based on a cross-time map structure.

[0005] A first aspect of this application provides a method for predicting unsteady flow fields at moving boundaries based on a cross-time-map structure, the method comprising: Acquire source time node data and target time node coordinates of the unsteady flow field of the moving boundary. The time corresponding to the target time node coordinates is adjacent to the time corresponding to the source time node data, and the time corresponding to the source time node data is the previous time and the time corresponding to the target time node coordinates is the next time. The source time node data includes source node coordinates and source node physical quantities. The boundary of the moving object is the boundary of the object surface that moves or deforms with time. Define the nodes corresponding to the source time node data as the source node set, and define the nodes corresponding to the target time node position as the target node set; A trans-temporal bipartite graph is constructed based on the source node set and the target node set. The trans-temporal bipartite graph includes a set of trans-temporal edges pointing from the source node to the target node. The trans-temporal edges are established based on the spatial proximity relationship between the source node and the target node. Construct simultaneous spatial edges of source nodes within the source node set, and construct simultaneous spatial edges of target nodes within the target node set; Construct the geometric identity features of each target node in the target node set, wherein the geometric identity features are used to characterize the relative geometric relationship between the target node and the boundary of the moving object; The node coordinates of the source node set, the node physical quantities of the source node set, the node coordinates of the target node set, the geometric identity features of the target node set, the cross-time bipartite graph, the simultaneous spatial edges of the source nodes and the simultaneous spatial edges of the target nodes are input into a pre-trained graph neural network model to obtain the predicted flow field physical quantities of the target node set output by the graph neural network model.

[0006] The first aspect of this application provides a method for predicting unsteady flow fields at moving boundaries based on a cross-time graph structure. The nodes corresponding to the source time node data are defined as a source node set, and the nodes corresponding to the target time node positions are defined as a target node set. A cross-time bipartite graph is constructed based on the source node set and the target node set. Simultaneous spatial edges of the source nodes are constructed within the source node set, and simultaneous spatial edges of the target nodes are constructed within the target node set. Geometric identity features of each target node in the target node set are constructed. The node coordinates of the source node set, the node physical quantities of the source node set, the node coordinates of the target node set, the geometric identity features of the target node set, the cross-time bipartite graph, the simultaneous spatial edges of the source nodes, and the simultaneous spatial edges of the target nodes are input into a pre-trained graph neural network model to obtain the predicted flow field physical quantities of the target node set, thereby improving the accuracy of dynamic grid flow field prediction.

[0007] To achieve the above objectives, a second aspect of this application provides a dynamic boundary unsteady flow field prediction system based on a cross-timemap structure, the system comprising: The acquisition module is used to acquire source time node data and target time node coordinates of the unsteady flow field of the moving boundary. The time corresponding to the target time node coordinates is adjacent to the time corresponding to the source time node data, and the time corresponding to the source time node data is the previous time and the time corresponding to the target time node coordinates is the next time. The source time node data includes source node coordinates and source node physical quantities. The boundary of the moving object is the boundary of the object surface that moves or deforms with time. The definition module is used to define the nodes corresponding to the source time node data as the source node set, and the nodes corresponding to the target time node position as the target node set; The first module is used to construct a cross-temporal bipartite graph based on the source node set and the target node set. The cross-temporal bipartite graph includes a set of cross-temporal edges pointing from the source node to the target node. The cross-temporal edges are established based on the spatial proximity relationship between the source node and the target node. The second module is used to construct simultaneous spatial edges of source nodes within the source node set and construct simultaneous spatial edges of target nodes within the target node set. The third module is used to construct the geometric identity features of each target node in the target node set, wherein the geometric identity features are used to characterize the relative geometric relationship between the target node and the boundary of the moving object; The prediction module is used to input the node coordinates of the source node set, the node physical quantities of the source node set, the node coordinates of the target node set, the geometric identity features of the target node set, the cross-time bipartite graph, the simultaneous spatial edges of the source nodes and the simultaneous spatial edges of the target nodes into a pre-trained graph neural network model, so as to obtain the predicted flow field physical quantities of the target node set output by the graph neural network model.

[0008] To achieve the above objectives, a third aspect of this application provides an electronic device, including: at least one processor and a memory for communicatively connecting to the at least one processor; the memory stores instructions executable by the at least one processor, which are executed by the at least one processor to enable the at least one processor to perform the above-described method for predicting unsteady flow fields based on a cross-timemap structure.

[0009] To achieve the above objectives, a fourth aspect of this application provides a computer-readable storage medium storing computer-executable instructions for causing a computer to execute the above-described method for predicting unsteady flow fields based on a cross-timemap structure.

[0010] It is understood that the beneficial effects of the second to fourth aspects compared with the related technologies are the same as the beneficial effects of the first aspect compared with the related technologies. Please refer to the relevant description in the first aspect above, which will not be repeated here. Attached Figure Description

[0011] The above and / or additional aspects and advantages of this application will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which: Figure 1This is a flowchart illustrating a method for predicting unsteady flow fields at moving boundaries based on a cross-time map structure, as provided in an embodiment of this application. Figure 2 This is a diagram of the unsteady flow field prediction network structure and training framework provided in the embodiments of this application; Figure 3 This is a schematic diagram of the mean absolute error (MAE), root mean square error (RMSE), and maximum absolute error (Max|err|) between the predicted flow field and the actual flow field provided in the embodiments of this application; Figure 4 This is a schematic diagram of the predicted flow field (left figure) and the actual flow field (right figure) of velocity component u, pressure p and velocity component v provided in the embodiments of this application; Figure 5 yes Figure 4 The diagram shows the error distribution between the predicted flow field and the actual flow field for the velocity component u, pressure p, and velocity component v. Figure 6 This is a schematic diagram of a dynamic boundary unsteady flow field prediction system based on a cross-time map structure provided in an embodiment of this application; Figure 7 This is a schematic diagram of the hardware structure of the electronic device provided in the embodiments of this application. Detailed Implementation

[0012] The embodiments of this application are described in detail below. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain this application, and should not be construed as limiting this application.

[0013] In the description of this application, the use of terms such as "first," "second," etc., is for the purpose of distinguishing technical features only and should not be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated or the order of the technical features indicated.

[0014] In the description of this application, it should be understood that the orientation descriptions, such as up, down, etc., are based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this application.

[0015] In the description of this application, it should be noted that, unless otherwise explicitly defined, terms such as "setup," "installation," and "connection" should be interpreted broadly, and those skilled in the art can reasonably determine the specific meaning of the above terms in this application in conjunction with the specific content of the technical solution.

[0016] The propulsion system of flexible biomimetic underwater vehicles typically involves continuous boundary deformation during operation. This motion induces complex unsteady flow fields, including boundary layer evolution, vortex shedding, wake development, and strongly nonlinear changes in pressure and velocity fields. To obtain high-precision flow field information, high-fidelity simulations using computational fluid dynamics methods are usually required.

[0017] However, high-fidelity computational fluid dynamics simulations are costly and involve large datasets, making them difficult to directly apply to parameter optimization, real-time control, or closed-loop design. Therefore, it is necessary to construct reduced-order models capable of rapidly predicting unsteady flow fields.

[0018] Existing convolutional neural network reduction models based on regular data typically assume that the data is located at fixed sampling points, making it difficult to directly handle moving boundaries and unstructured sampling points. While existing flow field prediction methods based on graph neural networks can handle irregular nodes, they still have a key problem under conditions of dynamic mesh deformation, reconstruction, resampling, or fixed spatial window truncation: the node indices of node sets at adjacent time steps cannot guarantee that they correspond to the same spatial location or the same spatial functional region, and therefore cannot directly establish stable cross-time information transmission relationships based on node indices.

[0019] Under the condition of continuous deformation of flexible boundaries, the node positions change constantly with the moving boundary and mesh reconstruction; at the same time, the sampling points within the fixed spatial window will also be locally replaced over time. If we directly assume that there is a stable correspondence between the same index nodes at adjacent times, it is easy to cause errors in the propagation of physical quantities, reducing the prediction accuracy near the boundary and in the wake region.

[0020] Based on this, this application provides a method and system for predicting unsteady flow fields at moving boundaries based on a cross-time graph structure. By constructing a cross-time bipartite graph, flow field nodes at adjacent times are associated under spatial proximity, and geometric identity features are constructed for each target node to characterize its relative geometric relationship with the moving boundary. This enables the correct propagation of physical quantities under moving grid conditions, effectively alleviates the node identity drift problem, improves the prediction accuracy near the boundary and in the wake region, and maintains high computational efficiency. This provides an efficient flow field prediction method for parameter optimization, real-time control, and closed-loop design of flexible moving boundary structures.

[0021] The method for predicting unsteady flow fields at moving boundaries based on a cross-time graph structure provided in this application is implemented using artificial intelligence technology, specifically involving the application of graph neural network models in flow field prediction tasks. A graph neural network model is a deep learning model capable of directly transmitting information and learning features on graph-structured data. It communicates messages through edges between nodes and can effectively process data in non-Euclidean spaces. In this application embodiment, the graph neural network model is applied to predicting unsteady flow fields at moving boundaries. By organizing the flow field nodes into a graph structure through a cross-time bipartite graph and simultaneous spatial edges, the model can correctly capture the spatiotemporal evolution of physical quantities under dynamic grid conditions.

[0022] The dynamic boundary unsteady flow field prediction method based on a cross-timemap structure provided in this application can be applied to various application scenarios. In one application scenario, the method provided in this application can be applied to the rapid prediction of the flow field of a flexible biomimetic underwater vehicle, providing real-time flow field information for the vehicle's motion control and attitude adjustment. In another application scenario, the method provided in this application can be applied to the aerodynamic characteristic analysis of wave-shaped airfoils, assisting engineers in quickly evaluating the flow field evolution under different airfoil parameters during the design phase. In yet another application scenario, the method provided in this application can be applied to the wake flow field prediction of a manta ray-inspired propulsion device, providing data support for optimizing propulsion efficiency and formulating vortex shelving control strategies. It is understood that the above application scenarios are merely illustrative examples, and the method provided in this application can also be applied to other scenarios involving the prediction of dynamic boundary unsteady flow fields; this application does not specifically limit these applications.

[0023] The dynamic boundary unsteady flow field prediction method based on a cross-timemap structure provided in this application can be executed by an electronic device, which can be a terminal device or a server. The terminal device can be a smartphone, tablet, laptop, desktop computer, or workstation, etc., and the server can be a standalone server or a server cluster. The electronic device is equipped with a graph neural network model, which can output predicted flow field physical quantities based on the input flow field data. It is understood that this application does not limit the specific form of the electronic device.

[0024] Reference Figure 1 This application provides a method for predicting unsteady flow fields at moving boundaries based on a cross-time map structure. The method includes the following steps S110 to S160.

[0025] Step S110: Obtain source time node data and target time node coordinates of the unsteady flow field of the moving boundary. The time corresponding to the target time node coordinates is adjacent to the time corresponding to the source time node data, and the time corresponding to the source time node data is the previous time, and the time corresponding to the target time node coordinates is the next time. The source time node data includes source node coordinates and source node physical quantities. The boundary of the moving object is the boundary of the object surface that moves or deforms with time.

[0026] In this step, a flow field node refers to a discrete sampling point in the flow field computational domain, with each node corresponding to a spatial location within the flow field. Node coordinates are used to identify the spatial location of the flow field node, and node physical quantities include fluid dynamic parameters such as velocity components, pressure, and vorticity. In one implementation, node coordinates and physical quantities at multiple consecutive time points can be obtained from the output of a high-fidelity computational fluid dynamics simulation; in another implementation, the corresponding data can be obtained from measurement data from flow field experiments or from a pre-established flow field database.

[0027] Specifically, the data of adjacent time steps of the unsteady flow field at the moving boundary are first obtained. The source time (previous time step) node data includes node coordinates and known physical quantities, while the target time (next time step) only requires node coordinates. The physical quantities at the target time node coordinates are to be predicted and are the objects to be predicted by the model. The moving boundary is the surface boundary of an object that moves or deforms over time, such as a flexible wave airfoil or a biomimetic propulsion structure.

[0028] For example, unsteady flow field data with moving boundaries generated by computational fluid dynamics methods are acquired. Moving boundaries include, but are not limited to, flexible wave-shaped airfoils, biomimetic fish propulsion structures, manta ray propulsion structures, or other underwater vehicle structures with continuously deformable boundaries. High-fidelity flow field data includes nodal coordinates, pressure, and velocity components at multiple consecutive time points.

[0029] For any time The set of flow field node locations and the corresponding set of physical quantities are represented as follows: ; ; in, Indicates the first n The set of node positions at each moment. This represents the set of physical quantities at the corresponding node. d For spatial dimensions, C Let be the number of channels for the physical quantity. For a two-dimensional incompressible flow field, the nodal physical quantity can be expressed as: ; in, For pressure, and These are velocity components. For a three-dimensional flow field, the nodal physical quantities can further include a third-dimensional velocity component.

[0030] This step obtains flow field data for the unsteady flow field at multiple consecutive time points at the moving boundary, providing a data foundation for constructing a cross-timescale diagram and predicting the flow field in subsequent steps. Since the flow field data includes nodal coordinates and physical quantities from multiple consecutive time points, it can fully describe the spatiotemporal evolution characteristics of the unsteady flow field at the moving boundary, thus providing sufficient information to accurately predict the flow field state at future time points.

[0031] Step S120: Define the nodes corresponding to the source time node data as the source node set, and define the nodes corresponding to the target time node position as the target node set.

[0032] In this step, the source time and the target time are two consecutive sampling times in the time series. The source nodes in the source node set carry known physical quantities and serve as the state input for flow field prediction; the node coordinates of the target node set are obtained or pre-given before prediction, and their corresponding flow field physical quantities are the quantities to be predicted, which are the output objects of flow field prediction. Therefore, the model uses the position and geometric information of the target nodes during the prediction process, but does not use the actual flow field physical quantities of the target nodes.

[0033] Specifically, the flow field node set is selected from a set of flow field nodes at multiple consecutive time points based on the time series, choosing a set of flow field nodes between two adjacent time points. In one implementation, the flow field node sets between two adjacent time points can be selected sequentially according to time order, for example, starting from time... and time In the two flow field node sets with an additional time step, the time step will be... The set of flow field nodes is used as the source node set, and the time... The set of flow field nodes with a time step is added as the target node set. For continuous multi-step prediction, this embodiment adopts an adjacent time step recursive method. Under the condition that the node position sequence of multiple subsequent time steps is given in advance, the target node predicted physical quantity obtained in the previous prediction step is used as the source node physical quantity in the next prediction step, and the predicted flow field at subsequent time steps is obtained by recursion.

[0034] For example, a fixed Euler sampling window is set in the physical space. A fixed Euler sampling window covers the moving body and its main wake flow region, and maintains its spatial position throughout the time series.

[0035] For each moment From falling into the fixed Euler sampling window Select from the flow field data within NA set of representative nodes is formed for training and prediction of the reduced-order model. This approach avoids the sampling region from drifting synchronously with the moving boundary, ensuring that the model always focuses on the evolution of the moving boundary flow field within a fixed physical region.

[0036] For two adjacent moments and ,Will The sampling nodes at time t are defined as the source node set, and the sampling nodes at time t are defined as the source node set. The sampling nodes at time t are defined as the target node set: ; ; Among them, the source node set Carrying known physical quantities Target node set Corresponding physical quantity The object to be predicted.

[0037] Because the node positions in a dynamic mesh flow field change over time, nodes with the same index at adjacent time points do not guarantee corresponding to the same spatial location or the same spatial functional region. Therefore, this application does not adopt a fixed node index correspondence relationship, but instead treats nodes at adjacent time points as two independent node sets and establishes a spatial relationship between them through a graph structure. Through this step, the flow field nodes at two adjacent time points are defined as the source node set and the target node set, respectively, clarifying the input and output objects for flow field prediction. The source node set provides known flow field state information, the pre-given target node coordinates determine the spatial location of the flow field to be predicted, and the flow field physical quantities corresponding to the target nodes are predicted by the graph neural network model, providing a clear division of node roles for subsequent construction of cross-time graph structures and message passing.

[0038] Step S130: Construct a cross-time bipartite graph based on the source node set and the target node set. The cross-time bipartite graph includes cross-time edges pointing from the source node to the target node. The cross-time edges are determined based on the spatial proximity relationship between the source node and the target node.

[0039] In this step, a cross-temporal bipartite graph is constructed based on the source node set and the target node set. This cross-temporal bipartite graph is used to establish the cross-temporal association between the source node and the target node. The cross-temporal bipartite graph is a graph structure consisting of two node sets, the source node set and the target node set, and the edges connecting the nodes in the two sets. Cross-temporal edges are edges pointing from the source node to the target node, representing the cross-temporal correspondence between the source and target nodes. Spatial proximity refers to the degree of closeness between the source node and the target node in terms of spatial location. A spatial proximity relationship is considered to exist when the spatial distance between the source node and the target node is less than or equal to a preset threshold.

[0040] Specifically, spatial proximity search is used to establish cross-temporal edges between the source node set and the target node set. For each target node, the source node set is searched for spatially neighboring source nodes, and cross-temporal edges are established between the searched source nodes and the target node. Spatial proximity search can employ either radius search or nearest neighbor search. Radius search involves searching all source nodes within a preset radius centered on the target node. Nearest neighbor search involves searching a preset number of source nodes that are closest to each target node. Through these search methods, a set of cross-temporal edges pointing from source nodes to target nodes can be established, forming a cross-temporal bipartite graph.

[0041] For example, for the source node set and target node set Construct a bipartite graph across time: ; in, This represents the set of edges that cross time points from the source node to the target node.

[0042] For each target node Search for its spatially nearest source node in the set of source nodes. When the source node... With the target node satisfy: ; Establish from source node i Point to target node j The time-crossing edge: ; in, The radius of the spatial search.

[0043] For the target node Its cross-time source node domain can be represented as: ; The corresponding set of edges across time points can be represented as: ; When the target node is within the radius If no source node exists in the target area, select several source nodes that are closest to the target node and connect to them to ensure that each target node can receive information from the source node.

[0044] Each cross-time edge carries relative geometric features: ; ; in, This represents the relative position vector between the source node and the target node. This represents the Euclidean distance between the two.

[0045] This step constructs a cross-temporal bipartite graph, establishing cross-temporal relationships between source and target nodes. Since cross-temporal edges are determined based on spatial proximity rather than simple node index correspondence, a cross-temporal information transmission relationship consistent with spatial proximity can be established, reducing physical quantity propagation deviations caused by node identity drift. The cross-temporal bipartite graph provides a topological foundation for cross-temporal message passing in graph neural network models, enabling the model to transmit physical quantity information from source nodes to spatially neighboring target nodes, thereby improving the prediction accuracy of unsteady flow fields at moving boundaries.

[0046] Step S140: Construct simultaneous spatial edges of the source nodes within the source node set, and construct simultaneous spatial edges of the target nodes within the target node set.

[0047] In this step, simultaneous spatial edges are constructed within the source node set and simultaneously within the target node set. Simultaneous spatial edges between source nodes refer to the edges between adjacent source nodes within the source node set, representing the local spatial structure of the flow field at the previous time step. Simultaneous spatial edges between target nodes refer to the edges between adjacent target nodes within the target node set, representing the local spatial structure of the flow field at the next time step. The construction of simultaneous spatial edges is based on the spatial proximity relationship between flow field nodes at the same time step; by establishing edges between spatially adjacent nodes, the local spatial topology of the flow field is formed.

[0048] Specifically, spatial proximity searches are performed on both the source node set and the target node set, establishing simultaneous spatial edges within each set. For the source node set, for each source node, other spatially neighboring source nodes are searched, and simultaneous spatial edges are established between them, forming a set of simultaneous spatial edges for the source nodes. For the target node set, other spatially neighboring target nodes are searched for for each target node, and simultaneous spatial edges are established between them, forming a set of simultaneous spatial edges for the target nodes. Spatial proximity searches can employ either K-nearest neighbor search or radius search. K-nearest neighbor search means that for each node, the K nearest neighbors are searched and edges are established. Radius search means that for each node, all nodes within a preset radius are searched and edges are established.

[0049] For example, within the source node set and the target node set, simultaneous spatial edges are constructed to describe the spatial correlation between local flow field nodes at the same time.

[0050] For the source node set Construct the set of internal spatial edges of the source node: ; For the target node set Construct the set of internal spatial edges of the target node: ; in, Indicates the source node Spatial neighborhood, Represents the target node The spatial neighborhood can be determined using K-nearest neighbor search or radius search.

[0051] For the internal spatial edge of the source node Its edge characteristics are determined by the source node. With neighboring source nodes The relative position vectors and Euclidean distance between them constitute: ; ; For the internal spatial edges of the target node Its edge features are determined by the target node. With neighboring target nodes The relative position vectors and Euclidean distance between them constitute: ; ; in, This represents the spatial edge characteristics inside the source node. This represents the spatial edge features within the target node. These edge features are used to describe the relative spatial relationships between adjacent nodes at the same time.

[0052] The internal spatial edges of the source node are used to extract the local spatial structure of the flow field at the previous time step; the internal spatial edges of the target node are used to enhance the spatial coordination between the prediction nodes at the target time step.

[0053] Therefore, the dynamic graph structure of adjacent time points can be represented as follows: ; in, This is a dynamic graph structure representing adjacent time points. For the source node set, For the target node set, Let f(x) be the set of cross-time edges pointing from the source node to the target node. Let the set of edges inside the source node be . Let be the set of edges inside the target node.

[0054] This step constructs simultaneous spatial edges within both the source and target node sets, forming local spatial topologies for each of the two time points. Simultaneous spatial edges at the source nodes enable the graph neural network model to transmit spatial information within the flow field of the previous time point, extracting local spatial features and facilitating the capture of local flow phenomena such as boundary layers and vortex structures. Simultaneous spatial edges at the target nodes enable the graph neural network model to transmit spatial information within the flow field of the subsequent time point, enhancing the spatial consistency of the target node prediction results and preventing unreasonable jumps in prediction values ​​between adjacent target nodes, thereby improving the overall smoothness and physical consistency of the flow field prediction results.

[0055] Step S150: Construct the geometric identity features of each target node in the target node set, wherein the geometric identity features are used to characterize the relative geometric relationship between the target node and the moving boundary.

[0056] In this step, a geometric identity feature is constructed for each target node. This geometric identity feature characterizes the relative geometric relationship between the target node and the moving boundary. The geometric identity feature is a feature vector composed of geometric relationship parameters between the target node and the moving boundary, used to provide identity information for the target node. The moving boundary refers to a flow field boundary that moves or deforms over time, such as the surface of a flexible airfoil or the outer surface of a biomimetic aircraft. The relative geometric relationship refers to the positional relationship of the target node relative to the moving boundary, including the distance from the target node to the moving boundary and the distance from the target node to a specific region of the moving boundary.

[0057] Specifically, geometric identity features are constructed by calculating various geometric relationship parameters between the target node and the moving boundary. In one implementation, the shortest distance from the target node to the moving boundary is calculated, reflecting the proximity between the target node and the moving boundary. In another implementation, the distance from the target node to a reference region at the head of the moving boundary is calculated, reflecting the position of the target node relative to the head of the moving boundary in the flow direction. In yet another implementation, the distance from the target node to a reference region at the tail of the moving boundary is calculated, reflecting the position of the target node relative to the tail of the moving boundary in the flow direction. These various geometric relationship parameters are combined into a feature vector, which serves as the geometric identity feature of the target node.

[0058] For example, to enhance the model's ability to identify fluid regions near moving boundaries, for each target node Constructing geometric identity features: ; in, This represents the shortest distance from the target node to the boundary of the target time movement. This represents the distance from the target node to the head reference point or head reference region of the moving object at the target time. It represents the distance from the target node to the tail reference point or tail reference region of the moving object at the target time.

[0059] The shortest distance from the target node to the motion boundary can be expressed as: ; in, express The boundary curve or boundary surface of motion at any given moment.

[0060] Head reference point and tail reference point Based on the motion boundary The anterior and posterior endpoints along the body axis or the main flow direction are determined and calculated: ; ; Based on the aforementioned geometric characteristics, the model can identify whether the target node is located in the near-wall region, the head-facing region, the tail-vortex shedding region, the wake region, or the far-field region, thereby improving the model's flow field prediction for the near-wall region, the head-facing region, or the wake region.

[0061] This step constructs a geometric identity feature for each target node, representing the relative geometric relationship between the target node and the moving boundary. Under moving mesh conditions, the spatial position of flow field nodes changes over time, making it difficult to stably identify the physical identity of nodes solely based on node coordinates. The geometric identity feature, by introducing the relative geometric relationship between the target node and the moving boundary, provides the target node with identity information associated with the moving boundary. This allows the graph neural network model to distinguish nodes located in different spatial functional regions based on the geometric identity feature, such as distinguishing nodes located within the boundary layer from nodes located in the wake region. This enables targeted feature learning and physical quantity prediction, improving the model's prediction accuracy near the boundary and in the wake region.

[0062] Step S160: Input the node coordinates and physical quantities of each source node in the source node set, the node coordinates and geometric identity features of each target node in the target node set, the cross-time bipartite graph, the simultaneous spatial edges of the source nodes and the simultaneous spatial edges of the target nodes into the pre-trained graph neural network model to obtain the predicted physical quantities of the unsteady flow field at each target node in the target node set.

[0063] In this step, the constructed graph structure data and node feature data are input into a pre-trained graph neural network model. The graph neural network model then outputs the predicted physical quantities at each target node in the target node set. The pre-trained graph neural network model refers to a graph neural network model that has flow field prediction capabilities after being trained on a training sample set. A graph neural network model is a deep learning model that can directly perform information transmission and feature learning on graph structure data. It updates node features by aggregating information from neighboring nodes through message passing between nodes. The predicted physical quantities refer to the predicted values ​​of the flow field physical quantities at the target nodes output by the graph neural network model, including velocity components, pressure, vorticity, etc.

[0064] Specifically, the node coordinates and physical quantities of the source nodes, and the node coordinates and geometric identity features of the target nodes are used as node feature inputs. The cross-temporal bipartite graph, the simultaneous spatial edges of the source nodes, and the simultaneous spatial edges of the target nodes are used as graph topology inputs, and all are fed into a pre-trained graph neural network model. The graph neural network model first encodes the physical quantities of the source nodes to obtain their initial latent features, and then encodes the node coordinates and geometric identity features of the target nodes to obtain their initial latent features. Next, the graph neural network model performs spatial message passing and feature updates within the source node set based on the simultaneous spatial edges of the source nodes. It also performs cross-temporal message passing and attention-weighted aggregation from the source nodes to the target nodes based on the cross-temporal bipartite graph, and performs spatial message passing and feature updates within the target node set based on the simultaneous spatial edges of the target nodes. Finally, the graph neural network model decodes the updated latent features of the target nodes to obtain the predicted physical quantities at each target node.

[0065] For example, such as Figure 2 As shown, the physical quantities of the source node, the position of the target node, the geometric identity features of the target node, the edge connection relationships and their edge features are input into the graph neural network encoding module (graph neural network model). Among them, the source node position and the target node position are used to construct the cross-time edge, the simultaneous spatial edge and its edge features.

[0066] The initial characteristics of the source node are determined by its physical quantities. The source node is constructed and its hidden features are obtained through the source node encoder: ; The initial features of the target node are determined by the target node's position. and geometric identity features The target node is constructed and its latent features are obtained through the target node encoder: ; For edges spanning different time periods, edges within the source node's internal space, and edges within the target node's internal space, the hidden features of the edges are obtained through the edge encoder: ; in, , , and These represent the source node encoder, target node encoder, and edge encoder, respectively.

[0067] Based on the internal space edges of the source node This involves simultaneous spatial message passing across the source node set. For the source node... Its neighborhood message can be represented as: ; The source node features are updated as follows: ; in, This indicates the source node's internal spatial edge message generation module. This indicates the source node feature update module.

[0068] Through this step, the hidden features of the source node not only contain its own physical quantity information, but also the local flow field structure information of its neighborhood.

[0069] Based on cross-time edge This transmits the source node information to the target node. For the target node... From the source node Transmit to the target node The cross-time message is represented as: ; in, This indicates the cross-time edge message generation module.

[0070] To enable the target node to adaptively distinguish the importance of different source nodes, attention-weighted aggregation is performed on messages from multiple source nodes connected to the same target node. For the target node The attention weights are represented as follows: ; ; in, This represents a trainable attention scoring function. Indicates the relationship with the target node A set of connected source nodes.

[0071] The hidden features of the target node are updated as follows: ; in, This indicates the feature update module across time steps.

[0072] This step is used to transfer the flow field information from the previous time step to the target node in the next time step, and adaptively select the main information source based on spatial proximity and the hidden feature state of the node.

[0073] After completing the cross-time information transmission, based on the internal spatial edges of the target node Spatial refinement is performed on the latent features of the target node after cross-time information transmission. For the target node... Its neighborhood message is represented as: ; The hidden features of the target node are further updated as follows: ; in, This represents the module for generating edge messages within the target node's internal space. This represents the target node feature update module.

[0074] This step is used to enhance the local spatial consistency between nodes predicting the target time, making the predicted physical quantities of adjacent nodes more continuous and coordinated.

[0075] The final latent features of the target node are input into the physical quantity decoder to obtain the predicted physical quantities at each target node at the target time: ; in, Represents a physical quantity decoder. Represents the target node exist The predicted physical quantity at a given time.

[0076] For a two-dimensional incompressible flow field, the prediction result is: ; For a three-dimensional incompressible flow field, the prediction result is: ; This yields the set of predicted flow field physical quantities for the entire set of target nodes: .

[0077] This step utilizes a pre-trained graph neural network model to predict the unsteady flow field at a moving boundary, obtaining the predicted physical quantities at each target node in the target node set. Because the graph neural network model performs information transmission and feature learning supported by a cross-time bipartite graph, simultaneous spatial edges, and geometric identity features, it can accurately capture the spatiotemporal evolution of physical quantities under moving mesh conditions, effectively mitigating the node identity drift problem and improving prediction accuracy near the boundary and in the wake region. Furthermore, the inference computation of the graph neural network model is far less than that of high-fidelity computational fluid dynamics simulation, meeting the response speed requirements of scenarios such as parameter optimization, real-time control, and closed-loop design.

[0078] Based on the above embodiments, this application further illustrates the construction process of a cross-temporal bipartite graph. In some embodiments, constructing a cross-temporal bipartite graph based on a source node set and a target node set includes the following steps.

[0079] Step S201: For each target node, search for source nodes within a preset radius in the source node set.

[0080] In this step, for each target node in the target node set, a search is conducted within the source node set using the target node's coordinates as the center and a preset radius as the search range. The preset radius refers to a pre-defined spatial search range used to determine whether a source node and a target node constitute a spatial proximity relationship. The value of the preset radius needs to be set based on the physical characteristics of the flow field and the grid scale. For example, it can be greater than the maximum displacement of flow field nodes between adjacent time points to ensure that enough source nodes can be found near the target node.

[0081] Specifically, spatial index data structures can be used to accelerate the search process. In one implementation, a spatial index structure such as a KD-tree or a ball tree can be constructed. All source nodes in the source node set are organized into the index structure according to their spatial coordinates. Then, a radius search is performed on each target node in the index structure to quickly find source nodes within a preset radius. In another implementation, a grid partitioning method can be used to divide the computational domain into multiple uniform grid cells. Source nodes are assigned to corresponding grid cells. Then, for each target node, only the source nodes in its own grid cell and adjacent grid cells are searched, thereby reducing the amount of computation required for the search.

[0082] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the preset radius can be set to twice the maximum displacement of the airfoil surface nodes between adjacent time points. For a target node located in the airfoil wake region, with the two-dimensional coordinates of the target node as the center and the preset radius as the search range, source nodes located within this circular range are searched in the source node set. Since the deformation of the airfoil between adjacent time points is small, multiple source nodes can usually be found near the target node. These source nodes are the candidate source nodes that have a spatial proximity relationship with the target node.

[0083] Step S202: If a source node is found within a preset radius, the source node is connected to the target node to form a cross-time edge of the target node.

[0084] In this step, it is determined whether a source node is found within a preset radius. If a source node is found, all the searched source nodes are connected to the target node, forming cross-time edges from the source nodes to the target node. These cross-time edges are directed edges, pointing from the source node to the target node, indicating that physical quantity information is transferred from the source node in the previous time step to the target node in the next time step.

[0085] Specifically, after finding source nodes within a preset radius, a cross-temporal edge pointing to the target node is created for each source node. This cross-temporal edge records the indices of the source and target nodes, as well as their relative positional relationship. The relative positional relationship can include the displacement vector from the source node to the target node, which can serve as the edge feature of the cross-temporal edge for subsequent message passing in the graph neural network model.

[0086] For example, for the target node located in the airfoil wake region, if three source nodes are found within a preset radius, three cross-time edges are established from these three source nodes to the target node. Each cross-time edge records the index of the corresponding source node, the index of the target node, and the displacement vector from the source node to the target node. Through the above operations, the target node establishes a cross-time association with the three source nodes.

[0087] Step S203: If no source node is found within the preset radius, select a preset number of source nodes that are closest to the target node from the source node set, and connect the selected source nodes to the target node to form a cross-time edge of the target node.

[0088] In this step, it is determined whether no source node is found within a preset radius. If no source node is found, a preset number of source nodes closest to the target node are selected from the source node set. These selected source nodes are then connected to the target node to form cross-time edges. The preset number refers to the number of nearest-neighbor source nodes set in advance, used to ensure that each target node has at least a certain number of cross-time edges, thereby ensuring that the graph neural network model can aggregate sufficient cross-time information for each target node. The value of the preset number needs to be set according to the grid density of the flow field and the prediction accuracy requirements.

[0089] Specifically, if no source node is found within a preset radius, all source nodes in the source node set are sorted according to their distance from the target node. A preset number of source nodes with the smallest distance are selected, and cross-time edges pointing to the target node are created for each of them. This situation typically occurs in regions where flow field nodes undergo large displacement movements, such as when significant airfoil deformation causes some target nodes to be without source nodes within a preset radius. In this case, the nearest neighbor search method ensures that these target nodes can still establish cross-time relationships.

[0090] For example, taking a two-dimensional flexible wave airfoil flow field as an example, when the airfoil undergoes a large pitching motion, some target nodes near the airfoil trailing edge may not be able to find source nodes within a preset radius. In this case, the distances between all source nodes and the target node are calculated in the source node set. The three source nodes closest to the target node are selected, and cross-time edges are established from these three source nodes to the target node. Through the above operations, even in regions where flow field nodes undergo large displacement motions, target nodes can establish cross-time relationships, ensuring that the information transmission of the graph neural network model covers all target nodes.

[0091] Through steps S201 to S203, cross-time edges are established for each target node, forming a complete cross-time bipartite graph. This cross-time bipartite graph employs a combination of radius search and nearest neighbor search. While ensuring that target nodes can find spatially neighboring source nodes, nearest neighbor search is used to supplement the search for special cases where no source node exists within a preset radius. This ensures that each target node has cross-time edges, thereby guaranteeing that the graph neural network model can aggregate cross-time information for all target nodes. This construction method can adapt to the non-uniformity of the spatial distribution of flow field nodes under dynamic grid conditions, establishing reasonable cross-time relationships in both dense and sparse regions of the flow field nodes, providing a reliable graph topology foundation for accurate prediction of unsteady flow fields at moving boundaries.

[0092] Based on the above embodiments, this application further describes the construction process of simultaneous spatial edges. In some embodiments, the simultaneous spatial edges of source nodes are constructed based on the source node set, and the simultaneous spatial edges of target nodes are constructed based on the target node set, including the following steps.

[0093] Step S301: Use K-nearest neighbor search or radius search to determine the spatial neighbors of each source node in the source node set and the spatial neighbors of each target node in the target node set, respectively.

[0094] In this step, either K-nearest neighbor search or radius search is used to determine the spatial neighbors of each source node in the source node set and each target node in the target node set, respectively. K-nearest neighbor search means that for each node, the K nearest neighbors in the set are searched as spatial neighbors, where K is a preset number of neighbors. Radius search means that for each node, all nodes within a preset radius in the set are searched as spatial neighbors. Spatial neighbors are nodes that are spatially adjacent to a given node, used to establish simultaneous spatial edges.

[0095] Specifically, spatial neighbor searches can be performed separately on the source node set and the target node set. For the source node set, each source node searches for its spatial neighbors within the source node set, resulting in a list of spatial neighbors for each source node. For the target node set, each target node searches for its spatial neighbors within the target node set, resulting in a list of spatial neighbors for each target node. When using the K-nearest neighbor search method, spatial index structures such as KD-trees can be constructed to accelerate the search process. When using the radius search method, grid partitioning can be employed to accelerate the search process.

[0096] For example, taking a two-dimensional flexible wave airfoil flow field as an example, a K-nearest neighbor search method is used, where K is set to 8. For each source node in the source node set, the 8 nearest source nodes in the source node set are searched as its spatial neighbors. Similarly, for each target node in the target node set, the 8 nearest target nodes in the target node set are searched as its spatial neighbors. Through the above operations, a list of spatial neighbors for each source node and each target node is obtained.

[0097] Step S302: Construct a set of internal spatial edges of the source nodes based on the spatial neighbors of each source node in the source node set, and construct a set of internal spatial edges of the target nodes based on the spatial neighbors of each target node in the target node set.

[0098] In this step, a set of internal spatial edges for the source node and a set of internal spatial edges for the target node are constructed based on the searched spatial neighbors. The set of internal spatial edges for the source node refers to the set of edges within the source node set determined by spatial neighbor relationships, used to characterize the local spatial structure of the flow field at the previous time step. The set of internal spatial edges for the target node refers to the set of edges within the target node set determined by spatial neighbor relationships, used to characterize the local spatial structure of the flow field at the next time step.

[0099] Specifically, for each source node in the source node set, a spatial edge is established between the source node and each of its spatial neighbors, and all spatial edges of the source nodes are aggregated to form the internal spatial edge set of the source node. Similarly, for each target node in the target node set, a spatial edge is established between the target node and each of its spatial neighbors, and all spatial edges of the target nodes are aggregated to form the internal spatial edge set of the target node. Each spatial edge can record the indices of its two endpoints and their relative positional relationship. The relative positional relationship can be used as an edge feature in the message passing process of the graph neural network model.

[0100] For example, for a source node in the source node set, if its spatial neighbor list contains 8 source nodes, then 8 internal spatial edges are established between the source node and each of these 8 spatial neighbors. After performing the above operation on all source nodes in the source node set, the set of internal spatial edges of the source nodes is obtained. Similarly, the same operation is performed on the target node set to obtain the set of internal spatial edges of the target nodes. These two sets of spatial edges represent the local spatial topology of the flow field at the previous and next time moments, respectively.

[0101] Through steps S301 to S302, the sets of internal spatial edges of the source node and the target node are constructed. The set of internal spatial edges of the source node enables the graph neural network model to perform spatial message passing in the flow field of the previous time step, extracting local spatial structural features of the flow field, which helps to capture the physical laws of local flow phenomena such as boundary layer evolution and vortex shedding. The set of internal spatial edges of the target node enables the graph neural network model to perform spatial message passing in the flow field of the next time step, enhancing the spatial consistency of the target node's prediction results, maintaining physical consistency in the prediction values ​​of adjacent target nodes, avoiding unreasonable spatial jumps, and thus improving the overall smoothness and accuracy of the flow field prediction results. Simultaneously, the moment-to-moment spatial edges and the cross-time bipartite graph together constitute a complete spatiotemporal graph structure, providing comprehensive topological support for the graph neural network model to perform spatiotemporal information passing and feature learning.

[0102] Based on the above embodiments, this application further describes the process of constructing geometric identity features. In some embodiments, constructing geometric identity features for each target node includes the following steps.

[0103] Step S401: Calculate the shortest distance from the target node to the moving boundary, and use it as the first geometric feature component.

[0104] In this step, the shortest distance from each target node to the moving boundary is calculated, and this shortest distance is used as the first geometric feature component. The shortest distance refers to the minimum distance among all points on the moving boundary surface from the target node, reflecting the proximity between the target node and the moving boundary. The moving boundary surface refers to the outer contour of the moving boundary, composed of a series of boundary points or boundary line segments. The first geometric feature component is a component of the geometric identity features, used to characterize the distance relationship between the target node and the moving boundary.

[0105] Specifically, for each target node, the distance from the target node to all boundary points on the moving boundary surface is calculated, and the minimum value is taken as the shortest distance. When the moving boundary surface is represented by discrete boundary points, all boundary points are traversed, the distance is calculated, and the minimum value is taken. When the moving boundary surface is represented by continuous boundary line segments or surfaces, analytical methods or numerical optimization methods can be used to calculate the shortest distance from the target node to the boundary line segments or surfaces. To accelerate the calculation process, a spatial index structure for the moving boundary surface can be constructed, such as a KD-tree or a bounding box hierarchy, to reduce the number of distance calculations.

[0106] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the moving boundary is the airfoil surface, represented by a series of discrete boundary points. For each target node, the distance from the target node to all boundary points on the airfoil surface is calculated, and the minimum value among them is taken as the shortest distance from the target node to the airfoil surface, i.e., the first geometric feature component. Target nodes located in the boundary layer region near the airfoil surface have a smaller first geometric feature component; target nodes located in the far-field region far from the airfoil surface have a larger first geometric feature component. Through the first geometric feature component, the graph neural network model can distinguish between nodes located in the boundary layer and nodes located in the far field, thereby performing targeted feature learning.

[0107] Step S402: Calculate the distance from the target node to the head reference region of the moving boundary, and use it as the second geometric feature component.

[0108] In this step, the distance from each target node to the moving boundary head reference region is calculated, and this distance is used as the second geometric feature component. The head reference region refers to the area of ​​the moving boundary that is forward in the flow direction, such as the head or leading edge region of an airfoil. The second geometric feature component reflects the position of the target node relative to the moving boundary head in the flow direction, and is used to characterize the relative positional relationship of the target node in the flow direction.

[0109] Specifically, the head reference region of the moving boundary is first determined. In one embodiment, the boundary point on the moving boundary surface that is furthest forward in the flow direction can be used as the head reference point, and the distance from the target node to this head reference point is calculated as the second geometric feature component. In another embodiment, the set of boundary points on the moving boundary surface that are within a predetermined range forward in the flow direction can be used as the head reference region, and the shortest distance from the target node to all boundary points within this region can be calculated as the second geometric feature component. In yet another embodiment, the difference between the coordinates of the target node in the flow direction and the coordinates of the head reference point in the flow direction can be calculated, and the absolute value of this difference can be used as the second geometric feature component.

[0110] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the reference point at the airfoil's nose is the airfoil's leading edge. For each target node, the distance from the target node to the airfoil's leading edge is calculated as the second geometric feature component. For target nodes located in the region in front of the airfoil's leading edge, the second geometric feature component reflects their proximity to the airfoil's leading edge; for target nodes located in the wake region behind the airfoil's trailing edge, the second geometric feature component reflects their distance from the airfoil's leading edge in the wake direction. Through the second geometric feature component, the graph neural network model can distinguish nodes located in different flow directions of the airfoil, which helps to capture the physical evolution of the flow field in the flow direction.

[0111] Step S403: Calculate the distance from the target node to the tail reference region of the moving boundary, and use it as the third geometric feature component.

[0112] In this step, the distance from each target node to the trailing reference region of the moving boundary is calculated, and this distance is used as the third geometric feature component. The trailing reference region refers to the area of ​​the moving boundary located towards the rear in the flow direction, such as the tail or trailing edge region of an airfoil. The third geometric feature component reflects the position of the target node relative to the trailing edge of the moving boundary in the flow direction, and is used to characterize the relative positional relationship between the target node and the wake region.

[0113] Specifically, the tail reference region of the moving boundary is first determined. In one embodiment, the boundary point on the moving boundary surface located at the rear of the flow direction can be used as the tail reference point, and the distance from the target node to this tail reference point is calculated as the third geometric feature component. In another embodiment, the set of boundary points on the moving boundary surface located within a predetermined range behind the flow direction can be used as the tail reference region, and the shortest distance from the target node to all boundary points within this region can be calculated as the third geometric feature component. In yet another embodiment, the difference between the coordinates of the target node in the flow direction and the coordinates of the tail reference point in the flow direction can be calculated, and the absolute value of this difference can be used as the third geometric feature component.

[0114] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the reference point at the tail of the airfoil is the trailing edge point. For each target node, the distance from the target node to the trailing edge point of the airfoil is calculated as the third geometric feature component. Target nodes located in the airfoil wake region have a smaller third geometric feature component, indicating that they are in the wake development region; target nodes located in the leading edge or far field region of the airfoil have a larger third geometric feature component. Through the third geometric feature component, the graph neural network model can identify nodes located in the wake region, which helps to capture the physical characteristics of flow phenomena such as vortex shedding and wake development.

[0115] Step S404: Combine the first geometric feature component, the second geometric feature component, and the third geometric feature component to obtain the geometric identity feature of the target node.

[0116] In this step, the calculated first, second, and third geometric feature components are combined to obtain the geometric identity features of the target node. The combination can be achieved by vector concatenation, where the three components are sequentially concatenated into a three-dimensional feature vector. The geometric identity features serve as the target node's identification information, providing a stable identity associated with the moving boundary under dynamic mesh conditions.

[0117] Specifically, the first, second, and third geometric feature components are concatenated into a feature vector in a predetermined order. In one implementation, the three scalar components can be directly concatenated into a three-dimensional vector. In another implementation, the three components can be normalized separately before concatenation to eliminate differences in dimensions and numerical ranges between different components. In yet another implementation, the three components can be nonlinearly transformed before concatenation, such as taking the logarithm of each component or performing a polynomial expansion, to enhance the expressive power of the geometric identity features.

[0118] For example, taking a two-dimensional flexible wave airfoil flow field as an example, for each target node, the first geometric feature component (the shortest distance to the airfoil surface), the second geometric feature component (the distance to the airfoil leading edge), and the third geometric feature component (the distance to the airfoil trailing edge) are concatenated into a three-dimensional feature vector, which serves as the geometric identity feature of the target node. This geometric identity feature describes the relative geometric relationship between the target node and the airfoil from multiple perspectives, providing rich node identity information for the graph neural network model.

[0119] Through steps S401 to S404, a geometric identity feature is constructed for each target node. This geometric identity feature consists of three components: the shortest distance from the target node to the moving boundary, the distance to the head reference region, and the distance to the tail reference region. It characterizes the relative geometric relationship between the target node and the moving boundary from multiple dimensions, including distance and flow direction. Under moving mesh conditions, the absolute coordinates of the flow field nodes change over time, but the relative geometric relationship between the node and the moving boundary more stably reflects the node's physical identity and spatial functional region. For example, nodes located within the boundary layer consistently have a smaller shortest distance to the moving boundary, while nodes located in the wake region exhibit specific distribution characteristics in their distance to the tail reference region. The graph neural network model utilizes this geometric identity feature to accurately distinguish nodes in different spatial functional regions, enabling targeted feature learning and physical quantity prediction, thereby effectively mitigating the node identity drift problem and improving prediction accuracy near the boundary and in the wake region.

[0120] Based on the above embodiments, this application further illustrates the process of obtaining predicted physical quantities from a graph neural network model. In some embodiments, the node coordinates and physical quantities of each source node in the source node set, the node coordinates and geometric identity features of each target node in the target node set, the cross-time bipartite graph, and the simultaneous spatial edges of the source nodes and the target nodes are input into a pre-trained graph neural network model to obtain the predicted physical quantities of the unsteady flow field at each target node in the target node set, including the following steps.

[0121] Step S501: Encode the physical quantities of each source node in the source node set to obtain the initial hidden features of each source node.

[0122] In this step, the graph neural network model encodes the physical quantities of each source node in the source node set, obtaining the initial latent features of each source node. Encoding refers to the process of mapping the physical quantities of the source nodes to a high-dimensional latent space, transforming the original physical quantities into high-dimensional feature vectors through a multilayer perceptron or linear transformation. The initial latent features are the feature representations of the source nodes before message passing within the graph neural network model, containing information about the physical quantities of the source nodes. These physical quantities include fluid dynamic parameters such as velocity components, pressure, and vorticity.

[0123] Specifically, the graph neural network model includes a source node feature encoder, which consists of multiple linear transformation layers and activation functions. For each source node, the encoder takes the physical quantity of that source node as input, performs multiple linear transformations and nonlinear activation processes, and outputs the initial latent features of that source node. In one implementation, the physical quantities of the source node include velocity components and pressure. The encoder concatenates the velocity components and pressure into an input vector, performs two linear transformations and ReLU activation functions, and outputs initial latent features of dimension D, where D is a preset latent feature dimension. In another implementation, the encoder can also take the node coordinates of the source node along with the physical quantity as input to introduce the spatial location information of the source node.

[0124] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the physical quantities of the source node include horizontal velocity components, vertical velocity components, and pressure values. The source node feature encoder of the graph neural network model concatenates these three physical quantities into a three-dimensional input vector. After the first linear transformation, the dimension is mapped from 3 to 64. After processing with the ReLU activation function, the dimension is mapped from 64 to 128 through the second linear transformation, resulting in an initial latent feature with a dimension of 128. This initial latent feature contains a high-dimensional representation of the source node physical quantities, providing a feature foundation for subsequent spatial message passing and cross-time message passing.

[0125] Step S502: Encode the node coordinates and geometric identity features of each target node in the target node set to obtain the initial latent features of each target node.

[0126] In this step, the graph neural network model encodes the node coordinates and geometric identity features of each target node in the target node set to obtain the initial latent features of each target node. The node coordinates and geometric identity features of the target nodes serve as encoding inputs, which are then processed by the encoder and converted into high-dimensional latent feature vectors. The initial latent features of the target nodes contain information about their spatial location and their relative geometric relationship with the moving boundary.

[0127] Specifically, the graph neural network model includes a target node feature encoder, which also consists of multiple linear transformation layers and activation functions. For each target node, the encoder concatenates the node coordinates and geometric identity features into an input vector. After multiple linear transformations and nonlinear activation processing, it outputs the initial latent features of the target node. Since the physical quantities of the target node are to be predicted and cannot be used as encoding input, the initial latent features of the target node are constructed solely based on the node coordinates and geometric identity features, without containing information about the physical quantities to be predicted, thus avoiding information leakage.

[0128] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the node coordinates of the target node include horizontal and vertical coordinates, and the geometric identity features include three components: the shortest distance to the airfoil surface, the distance to the leading edge of the airfoil, and the distance to the trailing edge of the airfoil. The target node feature encoder of the graph neural network model concatenates the node coordinates and geometric identity features into a five-dimensional input vector. After two layers of linear transformation and ReLU activation function processing, the output is an initial latent feature with a dimension of 128. This initial latent feature contains the spatial location information of the target node and its relative geometric relationship with the airfoil, providing a feature basis for subsequent cross-time message reception and spatial message transmission.

[0129] Step S503: Based on the simultaneous spatial edges of the source nodes, perform spatial message passing and feature updates within the source node set to obtain the updated hidden features of the source nodes.

[0130] In this step, the graph neural network model performs spatial message passing and feature updates within the set of source nodes based on the simultaneous spatial edges of the source nodes, resulting in updated latent features of the source nodes. Spatial message passing refers to the process by which the source node receives messages from its spatial neighbors via simultaneous spatial edges and fuses these messages with its own features. Feature updating refers to the source node updating its own latent features based on the received messages, ensuring that the updated latent features incorporate local spatial structure information.

[0131] Specifically, the graph neural network model includes a source node spatial message passing module. For each source node, this module determines its spatial neighbors through its simultaneous spatial edges and receives messages from each neighbor. These messages may include the latent features of neighboring nodes and their relative positions to the source node. The module aggregates all received messages using methods such as summation, averaging, or max pooling. Then, it fuses the aggregated messages with the current latent features of the source node using methods such as linear transformation after concatenation or gated fusion to obtain updated latent features. This spatial message passing and feature update process can be repeated multiple times to expand the receptive field.

[0132] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the source node spatial message passing module of the graph neural network model performs the following operations on each source node. For source node A, its eight spatial neighbors are determined through the spatial edges of the source node at the same moment. Messages are received from each spatial neighbor, including the latent features of the neighbor node and the displacement vector from the neighbor node to source node A. The module sums and aggregates the eight messages, then concatenates the aggregation result with the current latent features of source node A, and obtains the updated latent features of source node A after a linear transformation. This updated latent feature integrates the information of source node A and its spatial neighbors, containing the local spatial structure features of the flow field at the previous moment, which helps to capture local flow phenomena such as boundary layers and vortex structures.

[0133] Step S504: Based on the cross-time bipartite graph, cross-time message passing and attention-weighted aggregation are performed from the source node to the target node to obtain the updated target node latent features.

[0134] In this step, the graph neural network model performs cross-temporal message passing and attention-weighted aggregation from the source node to the target node based on a cross-temporal bipartite graph, obtaining the updated latent features of the target node. Cross-temporal message passing refers to the process by which the source node sends a message to the target node through cross-temporal edges in the cross-temporal bipartite graph. The message content includes the latent features of the source node and the relative positional relationship between the source and target nodes. Attention-weighted aggregation refers to the process by which the target node sums the received messages from multiple source nodes according to attention weights. These attention weights reflect the importance of different source nodes to the prediction result of the target node.

[0135] Specifically, the graph neural network model includes a cross-time-matter message passing module. For each target node, this module identifies all source nodes connected to it using a cross-time-matter bipartite graph and receives messages from each source node. Then, the module calculates the attention weight of each source node for the target node and sums the messages from all source nodes according to their attention weights to obtain an aggregated message. Finally, the module fuses the aggregated message with the target node's current latent features to obtain updated target node latent features. These updated latent features integrate cross-time-matter information from the source nodes and the target node's own spatial location information, providing rich feature representations for subsequent spatial message passing and physical quantity prediction.

[0136] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the cross-time message passing module of the graph neural network model performs the following operations for each target node. For target node B, three source nodes connected to it are determined through a cross-time bipartite graph. Messages are received from these three source nodes, each including the updated latent features of the source node and the displacement vector from the source node to target node B. The module calculates the attention weight of each source node to target node B, and the three messages are weighted and summed according to the attention weights to obtain an aggregated message. Then, the aggregated message is fused with the current latent features of target node B to obtain the updated latent features of target node B. These latent features contain cross-time information from the flow field at the previous time step and the spatial location information of target node B, providing a key feature basis for predicting the physical quantities at target node B.

[0137] Step S505: Based on the simultaneous spatial edges of the target nodes, perform spatial message passing and feature updates within the target node set to obtain the final hidden features of each target node.

[0138] In this step, the graph neural network model performs spatial message passing and feature updates within the target node set based on the simultaneous spatial edges of the target nodes, obtaining the final hidden features of each target node. The principle of spatial message passing at the target nodes is similar to that at the source nodes, but the object of this passing is the entire set of target nodes, aiming to enhance the spatial consistency of the prediction results. The final hidden features are the feature representations obtained after all message passing and feature updates at the target nodes, containing cross-time information and simultaneous spatial information, used for decoding to obtain the predicted physical quantities.

[0139] Specifically, the graph neural network model includes a target node spatial message passing module. For each target node, this module determines its spatial neighbors through its simultaneous spatial edges, receives messages from each neighbor, aggregates the received messages, and fuses them with the target node's current latent features to obtain updated target node latent features. This process can be repeated multiple times to expand the receptive field of the target nodes, allowing the final latent features to contain a wider range of spatial structural information. After multiple layers of spatial message passing, the final latent features of each target node are obtained.

[0140] For example, taking a two-dimensional flexible wave airfoil flow field as an example, the target node spatial message passing module of the graph neural network model performs the following operations on each target node. For target node B, its eight spatial neighbors are determined through the spatial edges of the target node at the same time. Messages are received from each spatial neighbor, and the eight messages are summed and aggregated before being fused with the current latent features of target node B to obtain the updated latent features of target node B. After this process is performed in two layers, the final latent features of target node B are obtained. This final latent feature integrates cross-time information and simultaneous spatial structure information, providing a comprehensive feature representation for accurately predicting the physical quantities at target node B.

[0141] Step S506: Decode the final hidden features of each target node to obtain the predicted physical quantities at each target node.

[0142] In this step, the graph neural network model decodes the final latent features of each target node to obtain the predicted physical quantities at each target node. Decoding refers to the process of mapping the final latent features of the target nodes from the high-dimensional latent space back to the physical quantity space. This is achieved by converting the high-dimensional feature vectors into predicted physical quantity values ​​through a multilayer perceptron or linear transformation. The predicted physical quantities include predicted values ​​of fluid dynamic parameters such as velocity components, pressure, and vorticity.

[0143] Specifically, the graph neural network model includes a feature decoder, which consists of multiple linear transformation layers and activation functions. For each target node, the decoder takes the final latent features of that target node as input, performs multiple linear transformations and nonlinear activations, and then outputs the predicted physical quantity at that target node. In one implementation, the output dimension of the decoder is consistent with the number of physical quantities to be predicted; for example, when the physical quantities to be predicted include horizontal velocity components, vertical velocity components, and pressure values, the decoder output dimension is 3. In another implementation, the decoder can set independent output branches for different types of physical quantities to adapt to the numerical range and distribution characteristics of different physical quantities.

[0144] For example, taking the flow field of a two-dimensional flexible wave airfoil as an example, the feature decoder of the graph neural network model decodes the final latent features of the target node B. The decoder takes the final latent features of dimension 128 as input, performs a first-level linear transformation to map the dimension from 128 to 64, processes it through the ReLU activation function, and then performs a second-level linear transformation to map the dimension from 64 to 3, outputting the predicted values ​​of the horizontal velocity component, the vertical velocity component, and the pressure at the target node B. After performing the above decoding operation on all target nodes, the graph neural network model outputs the predicted physical quantities at each target node in the target node set, constituting the flow field prediction result for the next time step.

[0145] Through steps S501 to S506, the graph neural network model completes the entire processing flow from input to output. This process sequentially executes source node feature encoding, target node feature encoding, source node spatial message passing, cross-time message passing, target node spatial message passing, and feature decoding. It correctly transmits the physical quantity information of the source node to the target node through the cross-time bipartite graph, while simultaneously enhancing the local spatial structure features of the flow field through simultaneous spatial edges. This process fully utilizes the topological and feature information provided by the cross-time bipartite graph, simultaneous spatial edges, and geometric identity features, accurately capturing the spatiotemporal evolution of physical quantities under moving mesh conditions, effectively mitigating the node identity drift problem, and improving the prediction accuracy of unsteady flow fields at moving boundaries.

[0146] Based on the above embodiments, this application further illustrates the attention-weighted aggregation process in cross-temporal message passing. In some embodiments, cross-temporal message passing and attention-weighted aggregation from the source node to the target node based on a cross-temporal bipartite graph includes the following steps.

[0147] Step S601: For each target node, calculate the attention weights of each source node connected to the target node for that target node.

[0148] In this step, the graph neural network model calculates the attention weights of each source node connected to the target node for each target node. The attention weight refers to the importance of a source node to the target node, reflecting the proportion of the source node's information contributing to the target node's feature update. The attention weight ranges from 0 to 1, and the sum of the attention weights of all source nodes connected to the same target node is 1.

[0149] Specifically, the graph neural network model includes an attention weight calculation module. For each target node, this module obtains all source nodes connected to it. For each source node, it calculates an attention score based on the latent features of the source node, the latent features of the target node, and their relative positional relationship. In one implementation, the attention score can be obtained by concatenating the latent features of the source nodes, the latent features of the target node, and the relative position vector, followed by a linear transformation and activation function processing. In another implementation, the attention score can be calculated using a dot product attention mechanism, i.e., performing a dot product operation between the latent features of the source nodes and the latent features of the target node, followed by scaling and activation processing. After obtaining the attention scores for all source nodes, the module normalizes the attention scores.

[0150] For example, taking a two-dimensional flexible wave airfoil flow field as an example, for the target node B, there are three source nodes connected to it, denoted as source node one, source node two, and source node three. The attention weight calculation module concatenates the latent features of source node one, the latent features of the target node B, and the displacement vector from source node one to the target node B into an input vector. After linear transformation and processing with the LeakyReLU activation function, the attention score of source node one is obtained. Similarly, the attention scores of source node two and source node three are calculated. Then, softmax normalization is performed on the three attention scores to obtain the attention weights of the three source nodes. If source node one is closest to the target node B, its attention weight is usually the largest, indicating that the information of source node one contributes the most to the feature update of the target node B.

[0151] Step S602: Weight the messages from each source node according to their corresponding attention weights to obtain weighted messages.

[0152] In this step, the graph neural network model weights the messages from each source node according to their corresponding attention weights to obtain a weighted message. The source node's message refers to the information sent by the source node to the target node through cross-time edges; the message content may include the source node's latent features and the relative positional relationship between the source and target nodes. The weighted message is the result obtained by multiplying the source node's message by its corresponding attention weight, reflecting the actual contribution of the source node's message to the target node's feature update.

[0153] Specifically, for each source node, the graph neural network model multiplies the source node's message vector by its attention weights to the target node to obtain the weighted message of that source node. In one implementation, the message vector may include the source node's latent features and relative position vector, and the weighted message is the result of multiplying each element of the message vector by the attention weights. In another implementation, the message vector may first fuse the source node's latent features and relative position vectors, and then multiply by the attention weights to obtain the weighted message.

[0154] For example, for the three source nodes of target node B, assume that the attention weight of source node 1 is 0.5, the attention weight of source node 2 is 0.3, and the attention weight of source node 3 is 0.2. The graph neural network model multiplies the message vector of source node 1 by 0.5 to obtain the weighted message of source node 1. Multiplying the message vector of source node 2 by 0.3 to obtain the weighted message of source node 2. Multiplying the message vector of source node 3 by 0.2 to obtain the weighted message of source node 3. Through weighting, the message of source node 1, which is closer to target node B, occupies a larger proportion in the aggregation result, while the message of source node 3, which is farther from target node B, occupies a smaller proportion in the aggregation result.

[0155] Step S603: Sum and aggregate the weighted messages of all source nodes connected to the same target node to obtain an aggregated message.

[0156] In this step, the graph neural network model sums and aggregates the weighted messages of all source nodes connected to the same target node to obtain an aggregated message. The aggregated message is the sum of the weighted messages of all source nodes, comprehensively reflecting the contribution of all source nodes to the feature update of the target node. Since the weighted messages have already been weighted according to attention weights, the aggregated message highlights the information of important source nodes and suppresses the information of secondary source nodes.

[0157] Specifically, for each target node, the graph neural network model sums the weighted messages of all source nodes connected to that target node element-wise to obtain the aggregated message of that target node. In one implementation, the aggregated message is a simple summation of all weighted messages. In another implementation, the aggregated message can be processed by a linear transformation and activation function after summation to enhance its expressive power.

[0158] For example, for target node B, the graph neural network model sums element-wise the weighted messages from source node one, source node two, and source node three to obtain the aggregated message of target node B. This aggregated message integrates the information from the three source nodes and is weighted according to attention weights, which can accurately reflect the influence of the flow field at the previous moment on the evolution of physical quantities at target node B.

[0159] Step S604: Perform feature fusion between the aggregated message and the initial latent features of the target node to obtain the updated latent features of the target node.

[0160] In this step, the graph neural network model fuses the aggregated message with the initial latent features of the target node to obtain the updated latent features of the target node. Feature fusion refers to the process of merging the aggregated message with the current latent features of the target node, so that the updated latent features simultaneously contain cross-time information from the source node and the spatial location information of the target node itself. The updated latent features of the target node will be used for subsequent target node spatial message passing or feature decoding.

[0161] Specifically, the graph neural network model fuses the aggregated message with the current latent features of the target node. In one implementation, the fusion method can employ a concatenation followed by a linear transformation, where the aggregated message and the current latent features are concatenated into a long vector, and then a linear transformation is performed to obtain the updated latent features. In another implementation, the fusion method can employ a gating mechanism, where a gating unit calculates the fusion ratio between the aggregated message and the current latent features, and then performs a weighted fusion according to this ratio. In yet another implementation, the fusion method can employ a residual connection, where the aggregated message is linearly transformed and then added to the current latent features to obtain the updated latent features.

[0162] For example, for target node B, the graph neural network model concatenates the aggregated message with the current latent features of target node B to obtain a concatenated vector. After a linear transformation and ReLU activation function, the updated latent features of target node B are obtained. This latent feature integrates cross-time information from the source node and the spatial location information of target node B itself, providing a rich feature representation for subsequent spatial message transmission and physical quantity prediction of the target node.

[0163] Through steps S601 to S604, the graph neural network model completes the attention-weighted aggregation process in cross-time message passing. This process differentiates the weighting of messages from different source nodes by calculating attention weights, ensuring that information from source nodes closer to the target node and with stronger physical connections occupies a larger proportion in the aggregation result, while information from source nodes farther from the target node and with weaker physical connections occupies a smaller proportion. This attention-weighted aggregation method can more accurately represent the information transmission relationship between source and target nodes based on spatial proximity, avoiding the physical quantity propagation deviation caused by simple average aggregation, thereby further improving the prediction accuracy of unsteady flow fields at moving boundaries. Especially near boundaries with large flow field gradients and in the wake region, the attention mechanism can effectively improve the accuracy of the prediction results.

[0164] Based on the above embodiments, this application further illustrates the training process of a graph neural network model. In some embodiments, the training process of a graph neural network model includes the following steps.

[0165] Step S701: Construct a training sample set and an initial graph neural network model.

[0166] In this step, the training samples contain complete graph structure information for adjacent time steps, and the training sample labels are the actual physical quantities of the training target node at the next time step. Specifically, each sample contains the coordinates of the training source node at the previous time step, the physical quantities of the training source node, the coordinates of the training target node at the next time step, the geometric identity features of the training target node, and three graph structures constructed based on these nodes: a training bipartite graph across time steps (connecting the training source node and the training target node), a simultaneous spatial edge of the training source node, and a simultaneous spatial edge of the training target node. Among them, the geometric identity features of the training target node are constructed based on the relative geometric relationship between the training target node and the boundary of the moving object at the next time step, and the training bipartite graph across time steps, the simultaneous spatial edges of the training source node, and the simultaneous spatial edges of the training target node are constructed based on the coordinates of the training source node and the coordinates of the training target node.

[0167] After receiving the above input, the graph neural network model performs feature encoding and message passing sequentially. First, the model encodes the physical quantities of the training source nodes, as well as the coordinates and geometric identity features of the training target node, to obtain their respective initial latent features. Then, it uses the simultaneous spatial edges of the training source nodes to perform spatial message passing, extracting the local structural information of the flow field from the previous time step. Next, it passes the information from the training source nodes to the training target node through a cross-time bipartite graph, and performs attention-weighted aggregation on the messages from multiple training source nodes connected to the same training target node. Finally, it uses the simultaneous spatial edges of the training target nodes to spatially refine the aggregated features, making the prediction results more continuous and consistent in space, and ultimately decodes and outputs the predicted physical quantities of the training target node.

[0168] The model training employs supervised learning, using the actual physical quantities of the training target nodes as the supervision signal. The training loss is defined as the reconstruction error between the model's predicted physical quantities and the actual physical quantities, and can be either mean absolute error or mean squared error. By minimizing this loss function, the backpropagation algorithm is used to perform end-to-end optimization of all trainable parameters in the model (including each encoder, message passing module, and decoder). After training, the model can quickly predict the distribution of flow field physical quantities at the next time step based on the graph structure input at any adjacent time steps.

[0169] For example, taking a two-dimensional flexible undulating airfoil flow field as an example, a training sample set is constructed from high-fidelity computational fluid dynamics simulation results. The simulation conditions include different airfoil oscillation frequencies, oscillation amplitudes, and incoming flow velocities, generating a total of 500 sets of simulation results. For each set of simulation results, flow field data from two adjacent time points are selected from multiple consecutive time points to construct a cross-time bipartite graph, simultaneous temporal spatial edges, and geometric identity features. The actual physical quantities of the later time point are used as labels to form the training samples. The final training sample set contains approximately 10,000 training samples, covering the flow field states at different time points and under different conditions. The initial graph neural network model is set to include a two-layer source node spatial message passing module, a one-layer cross-time message passing module, and a two-layer target node spatial message passing module, with the latent feature dimension set to 128.

[0170] Step S702: Input the training samples from the training sample set into the initial graph neural network model to obtain the predicted physical quantities at each target node.

[0171] In this step, the training samples from the training sample set are input into the initial graph neural network model, and the model outputs the predicted physical quantities at each target node. This process is consistent with the graph neural network model processing flow described in steps S501 to S506 above, except that an initial graph neural network model is used here, and its model parameters have not yet been optimized.

[0172] Specifically, training samples are selected from the training sample set. The source node coordinates and physical quantities, target node coordinates and geometric identity features, cross-time bipartite graph, and simultaneous spatial edges of the source node and target node are input into the initial graph neural network model. The initial graph neural network model undergoes feature encoding, spatial message passing, cross-time message passing, and feature decoding to output the predicted physical quantities at each target node. Batch training can be used, where multiple training samples are grouped into a batch and input into the model each time to improve training efficiency.

[0173] For example, a batch of 32 training samples is selected from the training sample set, and this batch of training sample data is input into the initial graph neural network model. The model performs forward propagation computation on each training sample, outputting the predicted physical quantities at the target nodes in each training sample. These predicted physical quantities are compared with the corresponding true physical quantity labels to calculate the reconstruction loss.

[0174] Step S703: Construct a reconstruction loss based on the predicted physical quantities and the real physical quantities in the training samples.

[0175] In this step, a reconstruction loss is constructed based on the predicted physical quantities and the actual physical quantities in the training samples. The reconstruction loss measures the difference between the predicted and actual physical quantities, guiding the optimization of the model parameters. The actual physical quantities refer to the actual values ​​of the flow field at the next time step in the training samples, serving as a reference standard for the prediction results.

[0176] Specifically, the difference between the predicted and actual physical quantities is calculated, and a reconstruction loss is constructed using an appropriate loss function. In one implementation, the loss function can be the mean squared error loss, which is the average of the squares of the differences between the predicted and actual physical quantities. In another implementation, the loss function can be the mean absolute error loss, which is the average of the absolute values ​​of the differences between the predicted and actual physical quantities. In yet another implementation, the loss function can be a weighted combination, assigning different weights to different types of physical quantities; for example, weighting velocity and pressure components separately and then summing them. Furthermore, a regularization term can be added to the loss function to prevent overfitting of the model.

[0177] For example, the mean squared error loss function is used to construct the reconstruction loss. For each training sample, the squares of the differences between the predicted velocity and pressure components and the actual velocity and pressure components are calculated. Then, the average of the squared differences over all target nodes and all physical quantity types is taken to obtain the reconstruction loss for that training sample. For a batch of 32 training samples, the reconstruction loss for each training sample is calculated, and the average value is taken as the reconstruction loss for that batch. This reconstruction loss reflects the accuracy of the initial graph neural network model in predicting the flow field under the current parameters; the smaller the loss value, the closer the model's prediction is to the true value.

[0178] Step S704: Optimize the model parameters of the initial graph neural network model with the goal of minimizing the reconstruction loss to obtain the trained graph neural network model.

[0179] In this step, the model parameters of the initial graph neural network model are optimized with the goal of minimizing the reconstruction loss, resulting in a trained graph neural network model. Model parameter optimization refers to calculating the gradient of the reconstruction loss with respect to the model parameters using the backpropagation algorithm and updating the model parameters according to the gradient descent direction, thereby gradually reducing the reconstruction loss. The trained graph neural network model is one whose reconstruction loss converges to a smaller value after multiple parameter optimizations, possessing the ability to accurately predict unsteady flow fields at moving boundaries.

[0180] Specifically, an optimization algorithm is used to iteratively optimize the model parameters. In one implementation, the optimization algorithm can be stochastic gradient descent or its variants, such as the Adam optimizer or the AdamW optimizer. The reconstruction loss is calculated in each training batch, and the gradient of the loss with respect to all model parameters is calculated using backpropagation. Then, the model parameters are updated according to the update rules of the optimization algorithm. This process is repeated multiple times on all training samples until the reconstruction loss converges to a preset threshold or the number of training epochs reaches a preset upper limit. During training, a learning rate decay strategy can be used, i.e., a larger learning rate is used in the early stages of training to accelerate convergence, and a smaller learning rate is used in the later stages to improve convergence accuracy. An early stopping strategy can also be used, i.e., training is stopped early when the loss on the validation set no longer decreases to prevent overfitting.

[0181] For example, the Adam optimizer is used to optimize the model parameters, with an initial learning rate set to 0.001. In each training epoch, the training sample set is divided into multiple batches, and each batch is sequentially input into the model for forward and backward propagation to update the model parameters. A total of 200 epochs are trained, and the model's predictive performance is evaluated on the validation set after each epoch. Training is stopped early when the loss on the validation set no longer decreases for 10 consecutive epochs, and the model parameters with the minimum loss on the validation set are saved as the trained graph neural network model. This trained model can accurately predict unsteady flow fields with moving boundaries and can be applied to practical tasks involving rapid flow field prediction.

[0182] In one embodiment, such as Figure 2 As shown, the set of physical quantities at the actual target time in computational fluid dynamics data. As a supervisory signal, the parameters of the graph neural network model are optimized. The training loss is defined as the set of predicted flow field physical quantities at the target time. With the set of physical quantities of the real flow field Reconstruction error between them.

[0183] In one implementation, the training loss uses the average L1 reconstruction error, which is expressed as: ; in, Represents the target node j exist The predicted physical quantity at time. Represents the target node j exist The real physical quantity at a given moment N Indicates the number of target nodes.

[0184] In another implementation, the training loss uses the mean squared reconstruction error, which is expressed as: ; in, Represents the target node j exist The predicted physical quantity at time. Represents the target node j exist The real physical quantity at a given moment N Indicates the number of target nodes.

[0185] By minimizing the above reconstruction loss, the trainable parameters in the source node encoder, target node encoder, side encoder, source node spatial feature extraction module at the same time step, cross-time information transmission module, target node spatial refinement module at the same time step, and physical quantity decoder are optimized.

[0186] After training, the graph neural network model can quickly predict the pressure and velocity components at the target node in the next moment based on the physical quantities of the source node in the previous moment, the positional relationship of the nodes in adjacent moments, the geometric identity features of the target node, and the dynamic graph structure.

[0187] After the model is trained, the set of source node positions at any given time is obtained. Source node physical quantity set Target node location set Cross-time edge Source node internal space edge Target node internal space edge and the geometric identity features of the target node By inputting the data into the trained graph neural network model, the predicted flow field at the target time can be obtained. .

[0188] The prediction results can be used for scenarios such as rapid reconstruction of flexible dynamic boundary flow fields, propulsion performance analysis of biomimetic underwater vehicles, optimization of motion parameters of undulating wings or flexible propulsion structures, dynamic boundary flow field proxy models, and closed-loop control simulation.

[0189] Through steps S701 to S704, the training process of the graph neural network model is completed. This training process constructs a training sample set covering various operating conditions, enabling the model to learn the spatiotemporal evolution of unsteady flow fields at moving boundaries under different conditions. By optimizing parameters with the goal of minimizing reconstruction loss, the model can gradually improve the prediction accuracy of flow field physical quantities. The trained graph neural network model fully utilizes the topological and feature information provided by the cross-time bipartite graph, simultaneous spatial edges, and geometric identity features. It can correctly capture the spatiotemporal evolution of physical quantities under moving mesh conditions, effectively alleviate the node identity drift problem, and provide a reliable model foundation for the rapid and accurate prediction of unsteady flow fields at moving boundaries.

[0190] To make the dynamic boundary unsteady flow field prediction method based on cross-time map structure provided in this application embodiment clearer, the following uses the flow field prediction of a two-dimensional flexible wave airfoil as an example to fully describe the method provided in this application embodiment. It is understood that this specific application example is only illustrative, and the method provided in this application embodiment can also be applied to other dynamic boundary unsteady flow field prediction scenarios.

[0191] This embodiment takes a two-dimensional flexible wave wing as the research object, uses the Fluent dynamic mesh method to generate high-fidelity data of the unsteady flow field of the dynamic boundary, and constructs a cross-time map structure based on the high-fidelity data to train a graph neural network reduced-order model, so as to realize the rapid prediction of pressure field and velocity field in the unsteady flow field induced by the two-dimensional wave wing.

[0192] Step 1: Acquisition of high-fidelity flow field data for a two-dimensional wave-shaped airfoil: In this embodiment, the characteristic length of the two-dimensional wave wing is set to The waveguide is in a tethered state, meaning it does not undergo overall translation or rotation; the unsteady fluid is induced only by the periodic vertical undulations of its flexible boundaries. The vertical displacement of the waveguide boundaries is controlled by a sinusoidal wave function, the expression of which is: ; in, x The coordinates are along the chord direction; Indicates the position of the wave wing boundary. x and time t Vertical displacement at the location; This is a function for smooth startup; The amplitude envelope function is distributed along the chord direction; k Wave number; ω is the angular frequency.

[0193] In this embodiment, the smooth start function is The amplitude envelope function is ; wave number is ,in The fluctuation frequency is The corresponding angular frequency is .

[0194] In Fluent dynamic mesh implementation, for any node on the wave wing boundary Its horizontal coordinates remain unchanged, while its vertical coordinates are updated incrementally. Let the node... The horizontal coordinate is And it remains unchanged during the dynamic mesh update process. Therefore, the node vertical coordinates are updated as follows: ; in, and Representing nodes respectively At any moment and The vertical coordinates. This incremental update method avoids the mesh deformation and discontinuity problem caused by directly overwriting the absolute position of the nodes.

[0195] At the same time, the overall velocity and angular velocity of the wave-shaped wing are both set to zero, that is: ; Therefore, the dynamic boundary flow field in this embodiment is induced solely by the flexible boundary fluctuations.

[0196] Based on the above definition of dynamic boundary, the Fluent dynamic mesh method is used to perform high-fidelity numerical simulation of a two-dimensional incompressible flow field. The computational domain is set to... The incoming flow velocity is The Reynolds number is The waveguide wall adopts a no-slip boundary condition, and its geometric position is updated through Fluent dynamic mesh UDF. During the calculation, the waveguide as a whole does not undergo rigid translation or rotation.

[0197] Time step set The total time step is set to 2000 steps, corresponding to a total physical time of The flow field data is saved at each time step during the calculation, and the saved physical quantities include pressure and two-dimensional velocity components. ,in p Indicates pressure, u Represents the horizontal velocity component. v This represents the vertical velocity component.

[0198] Step 2: Extract flow field data with a fixed Euler sampling window: To construct point cloud flow field data for training a graph neural network reduced-order model, this embodiment sets a fixed Eulerian sampling window in physical space. The fixed Euler sampling window covers the main body of the waveguide and its near-wake region, and remains spatially constant throughout the time series.

[0199] For each moment Extracting samples falling into a fixed Euler sampling window The calculation involves all computational grid points and their corresponding flow field physical quantities within the window. Since the moving grid deforms with the waveguide boundary, the number of computational grid points falling within the fixed Eulerian sampling window can vary at different times. In one specific implementation, the same number of N representative nodes are selected at each time point; in a general implementation, the number of sampling nodes can also vary with time. Therefore, the first... n The number of sampling nodes at each time point is denoted as . The corresponding set of node locations and the set of physical quantities are represented as follows: , ; in, Indicates the first n At the [time]th moment i The location of each sampling node This represents the pressure and velocity components at that node.

[0200] For two adjacent moments and Construct a single-step prediction sample: ; in, As the source time node location and physical quantity As input, the target time node position is given in advance. The real target flow field physical quantities are used as supervision labels only during the model training phase and are not used as inputs during the model prediction phase.

[0201] Step 3: Constructing the bipartite graph across time and the simultaneous spatial graph: For two adjacent moments and ,Will The set of sampling nodes within a fixed Euler sampling window is defined as the source node set: ; Will The target node set is defined as the sampling nodes within a fixed Euler sampling window at all times. ； In this context, the source node carries known physical quantities, while the target node carries physical quantities to be predicted.

[0202] In this embodiment, the cross-time edge is constructed using a radius search method. For any target node... In the source node set Search radius The nearest source node within, of which .

[0203] When the source node With the target node satisfy At that time, establish a system from the source node. Point to target node cross-time edge .

[0204] For each target node, the maximum number of neighbors across time steps is 16. If there are no source nodes within the radius, the node is backed up and one nearest source node is retained.

[0205] The edge features of each cross-time step consist of the relative position vector and the Euclidean distance: ; ; Within the source node set, the K-nearest neighbor method is used to construct the spatial edges of the source node at the same time, and the number of neighbors is... Set to 8; within the target node set, construct the spatial edges of the target node simultaneously using the K-nearest neighbor method, and the number of neighbors... Set it to 8.

[0206] This yields the set of edges inside the source node. and the set of internal space edges of the target node The internal spatial edges of the source node are used for extraction. The local spatial structure of the source flow field at any given time, with the internal spatial edges of the target node used for enhancement. Spatial consistency of prediction results for target nodes at any given time.

[0207] Therefore, in this embodiment, the dynamic graph structure at adjacent time points is represented as follows: ; Step 4: Constructing the geometric identity features of the target node: To enhance the model's ability to identify different flow regions near the waveguide, this embodiment constructs geometric identity features for each target node. For the target node... Its geometric identity features are represented as follows: ; in, Indicates the target node to The shortest distance at the boundary of the constantly fluctuating wing. This represents the distance from the target node to the leading edge reference point. This represents the distance from the target node to the trailing edge reference point.

[0208] like The wave wing boundary curve at time t is represented as: The shortest distance from the target node to the wave wing boundary is expressed as: ; The leading edge reference point and trailing edge reference point are respectively determined based on the wave fin boundary in the chord direction. x The minimum and maximum coordinates on the coordinate system are determined and calculated: ; ; The resulting geometric identity feature dimension is 3.

[0209] Step 5: Setting up the graph neural network model: This embodiment employs a three-stage graph neural network to construct a reduced-order prediction model. The model input includes the source node's physical quantities, the target node's location, the target's geometric identity features, cross-time step edge connections and their features, the source node's internal spatial edge connections and their features, and the target node's internal spatial edge connections and their features. The model output consists of the predicted physical quantities at each target node at the target time step. ; In this embodiment, the number of input physical quantity channels and the number of output physical quantity channels of the model are set to... coordinate dimension d =2, the geometric identity feature dimension of the target node is G=3, the cross-time edge feature dimension and the simultaneous spatial edge feature dimension are G=2, and the cross-time edge feature dimension and the simultaneous spatial edge feature dimension are G=3. The latent variable dimension of the graph neural network is H=128, and the number of spatial feature extraction layers at the source node at the same time step is... Number of information transmission layers across time periods The number of spatial refinement layers at the same time for the target node .

[0210] During the spatial feature extraction stage at the same time point of the source node, the model is based on Message passing is performed within the source node set to obtain the latent features of the source nodes that fuse local flow field structure information. During the cross-time information transmission phase, the model is based on... The flow field information from the source node is transmitted to the target node. During the spatial refinement phase at the target node at the same time, the model is based on... Local spatial fusion is performed on the latent features of the target node to obtain the final latent features of the target node used for decoding.

[0211] Finally, the physical quantity decoder outputs the pressure and velocity prediction results at the target node, thus obtaining the set of predicted flow field physical quantities within the fixed Euler sampling window at the target time: ; Step Six: Model Training Method This embodiment uses a single-step prediction method to construct training samples, that is, using... Timing node location and flow field physical quantities and The node position given in advance at a certain time is used as input, and the actual flow field physical quantities at that target time are used as the supervision target: ; in, The node positions are given in advance for the target time, and the actual flow field physical quantities at the target time are only used as training supervision labels.

[0212] The training and validation sets were divided in an 8:2 ratio. During training, the flow field physical quantities were... Normalization is performed, and the normalized statistics are calculated from the training set and applied to both the training and validation sets.

[0213] The model uses the Adam optimizer for parameter optimization, with a learning rate set to... The number of training rounds is set to 100, and the training loss is set to average. Reconstruction error: ; in, Represents the target node j The predicted physical quantity Represents the target node j The real physical quantity, This indicates the number of sampling nodes within the fixed Euler sampling window at the target time.

[0214] By minimizing the above reconstruction loss, the parameters in the source node encoder, target node encoder, side encoder, source node spatial feature extraction module at the same time step, cross-time information transmission module, target node spatial refinement module at the same time step, and physical quantity decoder are optimized.

[0215] Step 7: Rapid Flow Field Prediction Process After the model is trained, given a pre-defined set of target node locations, for any adjacent time points... and ,Will The set of source node positions within the constant Euler sampling window is fixed. Source node physical quantity set Target node location set Cross-time edges and their characteristics The internal spatial edges of the source node and their edge characteristics The internal spatial edges of the target node and their edge characteristics and the geometric identity features of the target node By inputting the trained graph neural network model, you can obtain... The set of predicted flow field physical quantities within a fixed Euler sampling window at all times: ; in, .

[0216] When performing continuous multi-step predictions, the sequence of node positions for multiple subsequent time steps can be given in advance, and the predicted flow field physical quantities obtained from the previous prediction step can be used as the source node physical quantities for the next prediction step, thus recursively obtaining the predicted flow field for subsequent time steps. The prediction results can be used for tasks such as rapid reconstruction of the flexible dynamic boundary flow field of a two-dimensional waveguide, near-wake flow analysis, propulsion direction force variation analysis, and waveguide motion parameter optimization.

[0217] This embodiment demonstrates that, in response to the problem that node positions in a two-dimensional wave-wing dynamic mesh flow field change over time and fixed index nodes no longer correspond to the same physical location, this application constructs a cross-time bipartite graph and a simultaneous spatial graph to transfer the flow field information from the previous time moment to the target node in the next time moment, and combines the geometric identity characteristics of the target node to achieve rapid prediction of the flow field near the moving boundary and the near-wake region.

[0218] like Figures 3-5 As shown, the predicted flow field is the flow field prediction result obtained by the prediction method of the present invention in this embodiment, and the real flow field is the simulated flow field result obtained by Fluent simulation calculation using dynamic mesh and overlapping mesh technology in this embodiment. Figure 3 The MAE, RMSE, and maximum absolute error Max|err| of the predicted flow field and the actual flow field in the continuous iteration steps of the rapid flow field prediction process in step seven of this embodiment are given respectively. It can be seen that the comparison results of different iteration steps are not significantly different, and the prediction results are accurate and stable. Figure 4 The three figures in the middle show the predicted flow field (left figure pred) and the actual flow field (right figure gt) for the velocity component u, pressure p, and velocity component v, respectively. Figure 5 The three rows of diagrams are given in sequence. Figure 4 Signed error distribution plots of the predicted and actual flow fields for the velocity component u, pressure p, and velocity component v, from... Figure 5 It can be seen that the signed errors of the predicted flow field and the actual flow field for velocity component u, pressure p and velocity component v are all near 0. The prediction results can reproduce the main flow structure near the wave wing and the wake region well. The errors are mainly concentrated in the region with large flow field gradient near the wall of the wave wing and in the wake.

[0219] like Figure 6 As shown in some embodiments of this application, a dynamic boundary unsteady flow field prediction system based on a cross-timemap structure is provided. The system includes an acquisition module 410, a definition module 420, a first module 430, a second module 440, a third module 450, and a prediction module 460. Specifically: The acquisition module 410 is used to acquire source time node data and target time node coordinates of the unsteady flow field of the moving boundary. The time corresponding to the target time node coordinates is adjacent to the time corresponding to the source time node data, and the time corresponding to the source time node data is the previous time, and the time corresponding to the target time node coordinates is the next time. The source time node data includes source node coordinates and source node physical quantities. The boundary of the moving object is the boundary of the object surface that moves or deforms with time. The definition module 420 is used to define the nodes corresponding to the source time node data as the source node set, and the nodes corresponding to the target time node position as the target node set; The first module 430 is used to construct a cross-time bipartite graph based on the source node set and the target node set. The cross-time bipartite graph includes a cross-time edge set from the source node to the target node. The cross-time edges are established based on the spatial proximity relationship between the source node and the target node. The second module 440 is used to construct simultaneous spatial edges of source nodes within the source node set and construct simultaneous spatial edges of target nodes within the target node set. The third module 450 is used to construct the geometric identity features of each target node in the target node set. The geometric identity features are used to characterize the relative geometric relationship between the target node and the boundary of the moving object. The prediction module 460 is used to input the node coordinates of the source node set, the node physical quantities of the source node set, the node coordinates of the target node set, the geometric identity features of the target node set, the cross-time bipartite graph, the simultaneous spatial edges of the source nodes and the simultaneous spatial edges of the target nodes into the pre-trained graph neural network model, so as to obtain the predicted flow field physical quantities of the target node set output by the graph neural network model.

[0220] It should be noted that the dynamic boundary unsteady flow field prediction system based on the cross-time map structure provided in this embodiment is based on the same inventive concept as the above-mentioned dynamic boundary unsteady flow field prediction method based on the cross-time map structure. Therefore, the relevant content of the above-mentioned dynamic boundary unsteady flow field prediction method based on the cross-time map structure is also applicable to the content of the dynamic boundary unsteady flow field prediction system based on the cross-time map structure. Therefore, it will not be repeated here.

[0221] The system defines the nodes corresponding to the source time node data as the source node set and the nodes corresponding to the target time node positions as the target node set. A cross-time bipartite graph is constructed based on the source and target node sets. Simultaneous spatial edges of the source nodes are constructed within the source node set, and simultaneous spatial edges of the target nodes are constructed within the target node set. Geometric identity features of each target node in the target node set are constructed. The node coordinates, node physical quantities, node coordinates, geometric identity features, cross-time bipartite graph, simultaneous spatial edges of the source and target nodes are input into a pre-trained graph neural network model to obtain the predicted flow field physical quantities of the target node set, thereby improving the accuracy of dynamic mesh flow field prediction.

[0222] This application also provides an electronic device, which includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the above-described method for predicting unsteady flow fields based on a cross-time map structure.

[0223] like Figure 7 , Figure 7 This is a schematic diagram of the hardware structure of an electronic device provided in an embodiment of this application. The electronic device includes: At least one memory; At least one processor; At least one program; The program is stored in memory, and the processor executes at least one program to implement the above-described method for predicting unsteady flow fields based on a cross-timemap structure of moving boundaries, as described in this disclosure.

[0224] This electronic device can be any smart terminal, including mobile phones, tablets, personal digital assistants (PDAs), and in-vehicle computers.

[0225] The electronic devices according to embodiments of this application will now be described in detail.

[0226] The processor 1600 can be implemented using a general-purpose central processing unit (CPU), microprocessor, application-specific integrated circuit (ASIC), or one or more integrated circuits, and is used to execute relevant programs to implement the technical solutions provided in the embodiments of this disclosure. The memory 1700 can be implemented as a read-only memory (ROM), static storage device, dynamic storage device, or random access memory (RAM). The memory 1700 can store the operating system and other application programs. When the technical solutions provided in the embodiments of this specification are implemented through software or firmware, the relevant program code is stored in the memory 1700 and is called and executed by the processor 1600 to execute the dynamic boundary unsteady flow field prediction method based on a cross-timemap structure according to an embodiment of this disclosure.

[0227] The input / output interface 1800 is used to implement information input and output. The communication interface 1900 is used to enable communication and interaction between this device and other devices. Communication can be achieved through wired means (such as USB, Ethernet cable, etc.) or wireless means (such as mobile network, WIFI, Bluetooth, etc.). Bus 2000 transmits information between various components of the device (e.g., processor 1600, memory 1700, input / output interface 1800, and communication interface 1900); The processor 1600, memory 1700, input / output interface 1800 and communication interface 1900 are connected to each other within the device via bus 2000.

[0228] This disclosure also provides a storage medium, which is a computer-readable storage medium storing computer-executable instructions for causing a computer to execute the above-described method for predicting unsteady flow fields based on a cross-timemap structure.

[0229] Memory, as a non-transitory computer-readable storage medium, can be used to store non-transitory software programs and non-transitory computer-executable programs. Furthermore, memory may include high-speed random access memory, and may also include non-transitory memory, such as at least one disk storage device, flash memory device, or other non-transitory solid-state storage device. In some embodiments, memory may optionally include memory remotely located relative to the processor, and these remote memories can be connected to the processor via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.

[0230] The embodiments described in this disclosure are for the purpose of more clearly illustrating the technical solutions of this disclosure and do not constitute a limitation on the technical solutions provided by this disclosure. As those skilled in the art will know, with the evolution of technology and the emergence of new application scenarios, the technical solutions provided by this disclosure are also applicable to similar technical problems.

[0231] Those skilled in the art will understand that the technical solutions shown in the figures do not constitute a limitation on the embodiments of this disclosure, and may include more or fewer steps than shown, or combine certain steps, or different steps.

[0232] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs.

[0233] Those skilled in the art will understand that all or some of the steps in the methods disclosed above, as well as the functional modules / units in the systems and devices, can be implemented as software, firmware, hardware, or suitable combinations thereof.

[0234] The terms “first,” “second,” “third,” “fourth,” etc. (if present) in the specification and accompanying drawings of this application are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of this application described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms “comprising” and “having,” and any related variations, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0235] It should be understood that in this application, "at least one (item)" means one or more, and "more than" means two or more. "And / or" is used to describe the relationship between related objects, indicating that three relationships can exist. For example, "A and / or B" can represent three cases: only A exists, only B exists, and both A and B exist simultaneously, where A and B can be singular or plural. The character " / " generally indicates that the preceding and following related objects are in an "or" relationship. "At least one (item) of the following" or similar expressions refer to any combination of these items, including any combination of single or plural items. For example, at least one (item) of a, b, or c can represent: a, b, c, "a and b", "a and c", "b and c", or "a and b and c", where a, b, and c can be single or multiple.

[0236] In the several embodiments provided in this application, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative; for instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some interfaces; the indirect coupling or communication connection between apparatuses or units may be electrical, mechanical, or other forms.

[0237] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.

[0238] Furthermore, the functional units in the various embodiments of this application can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0239] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes multiple instructions to cause an electronic device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of this application. The aforementioned storage medium includes various media capable of storing programs, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0240] The above is a detailed description of the preferred embodiments of this application. However, the embodiments of this application are not limited to the above-described implementation methods. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the embodiments of this application. All such equivalent modifications or substitutions are included within the scope defined by the claims of the embodiments of this application.

[0241] The embodiments of this application have been described in detail above with reference to the accompanying drawings. However, this application is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of this application.

Claims

1. A method for predicting unsteady flow fields at moving boundaries based on a cross-time-map structure, characterized in that, The method includes: Acquire source time node data and target time node coordinates of the unsteady flow field of the moving boundary. The time corresponding to the target time node coordinates is adjacent to the time corresponding to the source time node data, and the time corresponding to the source time node data is the previous time and the time corresponding to the target time node coordinates is the next time. The source time node data includes source node coordinates and source node physical quantities. The boundary of the moving object is the boundary of the object surface that moves or deforms with time. Define the nodes corresponding to the source time node data as the source node set, and define the nodes corresponding to the target time node position as the target node set; A trans-temporal bipartite graph is constructed based on the source node set and the target node set. The trans-temporal bipartite graph includes a set of trans-temporal edges pointing from the source node to the target node. The trans-temporal edges are established based on the spatial proximity relationship between the source node and the target node. Construct simultaneous spatial edges for source nodes within the source node set, and construct simultaneous spatial edges for target nodes within the target node set; Construct the geometric identity features of each target node in the target node set, wherein the geometric identity features are used to characterize the relative geometric relationship between the target node and the boundary of the moving object; The node coordinates of the source node set, the node physical quantities of the source node set, the node coordinates of the target node set, the geometric identity features of the target node set, the cross-time bipartite graph, the simultaneous spatial edges of the source nodes and the simultaneous spatial edges of the target nodes are input into a pre-trained graph neural network model to obtain the predicted flow field physical quantities of the target node set output by the graph neural network model.

2. The method for predicting unsteady flow fields at moving boundaries based on a cross-time diagram structure according to claim 1, characterized in that, The construction of a cross-time bipartite graph based on the source node set and the target node set includes: For each target node in the target node set, a source node with a spatial distance less than or equal to a preset radius is searched in the source node set, so as to connect the searched source node with the target node to form the cross-time edge; The step of connecting the searched source node with the target node to form the cross-time edge includes: If there is a source node in the set of source nodes whose spatial distance to the target node is less than or equal to the preset radius, determine the neighboring source nodes of the target node to establish a cross-time edge from the neighboring source node to the target node; If there is no source node in the source node set whose spatial distance to the target node is less than or equal to the preset radius, a preset number of source nodes closest to the target node are selected from the source node set and connected to the target node to form the cross-time edge.

3. The method for predicting unsteady flow fields at moving boundaries based on a cross-time diagram structure according to claim 1, characterized in that, The step of constructing simultaneous spatial edges of source nodes within the source node set and simultaneous spatial edges of target nodes within the target node set includes: The spatial neighborhood of the source node and the spatial neighborhood of the target node are determined by K-nearest neighbor search or radius search. A set of internal spatial edges of the source node is constructed based on the spatial neighborhood of the source node, and a set of internal spatial edges of the target node is constructed based on the spatial neighborhood of the target node. The source node's internal spatial edges are used to extract the local spatial structure of the flow field at the previous moment, while the target node's internal spatial edges are used to enhance the spatial coordination between the prediction nodes at the target moment.

4. The method for predicting unsteady flow fields at moving boundaries based on a cross-time diagram structure according to claim 1, characterized in that, The construction of the geometric identity features of each target node in the target node set includes: For each target node in the target node set, calculate the shortest distance from the target node to the boundary of the moving object to obtain the first geometric feature component; Calculate the distance from the target node to the head reference region of the moving object to obtain the second geometric feature component; Calculate the distance from the target node to the tail reference region of the moving object to obtain the third geometric feature component; The first geometric feature component, the second geometric feature component, and the third geometric feature component are combined to form the geometric identity feature of the target node.

5. The method for predicting unsteady flow fields at moving boundaries based on a cross-time diagram structure according to claim 1, characterized in that, The graph neural network model obtains the predicted flow field physical quantities of the target node set through the following steps: The node physical quantities of each source node in the source node set are encoded to obtain the initial hidden features of each source node. The node coordinates and geometric identity features of each target node in the target node set are encoded to obtain the initial latent features of each target node; Based on the simultaneous spatial edges of the source nodes, spatial message passing and feature updating are performed on the initial latent features of each source node in the set of source nodes to obtain the updated latent features of each source node. Based on the cross-time edge, the updated latent features of the source node are passed to the target node, and attention-weighted aggregation is performed on the passed messages of each source node connected to the same target node, so as to update the initial latent features of the target node according to the aggregation result, and obtain the cross-time updated latent features of each target node. Based on the simultaneous spatial edges of the target nodes, spatial message passing and feature updates are performed on the cross-time-lapse latent features of the target nodes to obtain the final latent features of each target node. The final hidden features of each target node are decoded to obtain the predicted flow field physical quantities of each target node.

6. The method for predicting unsteady flow fields at moving boundaries based on a cross-time diagram structure according to claim 5, characterized in that, The attention-weighted aggregation of messages transmitted by each source node connected to the same target node, to update the initial latent features of the target node based on the aggregation result, and to obtain the cross-time-lapse updated latent features of each target node, includes: For each source node connected to the same target node, calculate the attention weight of the source node to the target node based on the updated latent features of the source node and the initial latent features of the target node; The transmitted messages of the source node are weighted based on the attention weights to obtain a weighted message; The aggregation result is obtained by summing the weighted messages of each source node connected to the same target node; Based on the initial latent features of the target node and the aggregation result, the initial latent features of the target node are updated through feature fusion to obtain the cross-time updated latent features of the target node.

7. The method for predicting unsteady flow fields at moving boundaries based on a cross-time diagram structure according to claim 1, characterized in that, The graph neural network model is trained through the following steps: A training sample set and an initial graph neural network model are constructed. The training sample set includes flow field data training samples and training sample labels. The flow field data training samples include training node data at adjacent time steps. The training node data at adjacent time steps includes the coordinates of the training source node at the previous time step, the physical quantity of the training source node, the coordinates of the training target node at the next time step, the geometric identity features of the training target node, the training cross-time bipartite graph, the spatial edges of the training source node at the same time step, and the spatial edges of the training target node at the same time step. The training sample label is the actual physical quantity of the training target node at the next time step. The training sample set is input into the initial model of the graph neural network to obtain the target node predicted physical quantity output by the initial model of the graph neural network. A reconstruction loss is constructed based on the predicted physical quantities of the target node and the actual physical quantities of the target node. The initial model of the graph neural network is then optimized by minimizing the reconstruction loss to obtain the graph neural network model.

8. A predictive system for unsteady flow fields at moving boundaries based on a cross-time-map structure, characterized in that, The system includes: The acquisition module is used to acquire source time node data and target time node coordinates of the unsteady flow field of the moving boundary. The time corresponding to the target time node coordinates is adjacent to the time corresponding to the source time node data, and the time corresponding to the source time node data is the previous time and the time corresponding to the target time node coordinates is the next time. The source time node data includes source node coordinates and source node physical quantities. The boundary of the moving object is the boundary of the object surface that moves or deforms with time. The definition module is used to define the nodes corresponding to the source time node data as the source node set, and the nodes corresponding to the target time node position as the target node set; The first module is used to construct a cross-temporal bipartite graph based on the source node set and the target node set. The cross-temporal bipartite graph includes a set of cross-temporal edges pointing from the source node to the target node. The cross-temporal edges are established based on the spatial proximity relationship between the source node and the target node. The second module is used to construct simultaneous spatial edges of source nodes within the source node set and construct simultaneous spatial edges of target nodes within the target node set. The third module is used to construct the geometric identity features of each target node in the target node set, wherein the geometric identity features are used to characterize the relative geometric relationship between the target node and the boundary of the moving object; The prediction module is used to input the node coordinates of the source node set, the node physical quantities of the source node set, the node coordinates of the target node set, the geometric identity features of the target node set, the cross-time bipartite graph, the simultaneous spatial edges of the source nodes and the simultaneous spatial edges of the target nodes into a pre-trained graph neural network model, so as to obtain the predicted flow field physical quantities of the target node set output by the graph neural network model.

9. An electronic device, characterized in that, It includes at least one processor and a memory for communicatively connecting to the at least one processor; the memory stores instructions executable by the at least one processor, which, when executed by the at least one processor, enable the at least one processor to perform a dynamic boundary unsteady flow field prediction method based on a cross-timemap structure as described in any one of claims 1 to 7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer-executable instructions for causing a computer to perform a method for predicting unsteady flow fields based on a cross-timemap structure, as described in any one of claims 1 to 7.

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