Method, device, equipment, medium and program product for training of physical field neural network with cross-scale separable grid representation

Abstract

This application provides a method, apparatus, device, medium, and program product for training a physical field neural network with a cross-scale separable grid representation. The method includes: inputting a sample set of aircraft wing shapes into a first neural network; assigning M resolutions to M grid basis functions to balance global and local feature learning; capturing common features among samples through the weights of the first network; determining M specific fusion coefficients for each sample and processing them through the M grid basis functions to obtain M grid basis function features; fusing the grid basis function features according to the fusion coefficients to obtain sample-specific modulation features; obtaining predicted features based on the modulation features and common features and reconstructing the sample; and updating the parameter values ​​of the first network weights and grid basis functions based on the difference between the reconstructed result and the sample. This application can improve the parameter efficiency and model expressive power of physical field modeling.

Description

Technical Field

[0001] This application relates to the fields of physical field modeling and deep learning technology, and in particular to a method, apparatus, device, medium and program product for training physical field neural networks with cross-scale separable grid representation. Background Technology

[0002] Partial differential equations (PDEs) are a core tool for describing physical phenomena in many engineering and scientific fields, such as fluid mechanics and structural mechanics. In recent years, using neural networks for physical field modeling and PDE solving has become a research hotspot. To enhance the network's ability to characterize spatial structures, related technologies have introduced grid-based spatial modulation mechanisms. By constructing learnable grid features in the spatial domain and fusing them with coordinate input, the model's ability to represent local regions is improved.

[0003] However, the relevant methods still have the following problems when dealing with physical field modeling tasks such as aircraft wing shape representation. First, traditional single-scale grid structures cannot simultaneously take into account the global trend of the physical field and local high-frequency details. In aircraft wing shape representation, the overall wing profile and large-scale changes in the wingspan direction coexist with the drastic curvature changes in local areas such as the wingtip and leading edge in the same spatial domain. Single-scale grids mix high-frequency and low-frequency components in the same scale, making it difficult for the model to simultaneously characterize macroscopic topological structures and microscopic geometric features. Second, in the relevant technologies, the parameters of the grid basis functions and the neural network weights are independent of each other, and the two do not form an effective linkage during training. The grid basis functions cannot adaptively adjust their spatial representation method according to the specific modeling task, affecting the model's representation efficiency and reconstruction accuracy.

[0004] Therefore, how to construct a spatial representation mechanism that can simultaneously take into account global trends and local details, and achieve collaborative optimization of mesh parameters and network parameters, is a technical problem that urgently needs to be solved in current physical field neural modeling tasks. Summary of the Invention

[0005] The purpose of this application is to provide a method, apparatus, device, medium, and program product for training physical field neural networks with cross-scale separable grid representation, which can improve parameter efficiency and model expressive power in physical field modeling tasks.

[0006] To solve the above-mentioned technical problems, this application is implemented as follows: A first aspect of this application discloses a method for training a physical field neural network with a cross-scale separable grid representation, the method comprising: Input the set of aircraft wing shape samples into the first neural network to be trained; M resolutions are assigned to the M grid basis functions of the first neural network, with one grid basis function corresponding to one resolution; the M resolutions are used to balance global feature learning and local feature learning on the aircraft wing shape sample set; For the aircraft wing shape sample set, the common features among the various aircraft wing shape samples in the aircraft wing shape sample set are captured through the first network weights of the first neural network; For each aircraft wing shape sample, the first neural network determines M fusion coefficients specific to that aircraft wing shape sample. The aircraft wing shape sample is processed using the M grid basis functions at M different resolutions to obtain the M grid basis function features of the aircraft wing shape sample. Based on the M fusion coefficients specific to the aircraft wing shape sample, the M grid basis function features of the aircraft wing shape sample are fused to obtain the modulation features specific to the aircraft wing shape sample. Based on the modulation features specific to the aircraft wing shape sample and the common features, the predicted features of the aircraft wing shape sample are obtained; The predicted features of each aircraft wing shape sample are used to reconstruct the aircraft wing shape sample; Based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample, the values ​​of the first network weights of the first neural network and the parameter values ​​of the M grid basis functions are updated.

[0007] Optionally, for each aircraft wing shape sample, M fusion coefficients specific to that aircraft wing shape sample are determined by the first neural network, including: For each aircraft wing shape sample, latent variables specific to that aircraft wing shape sample are generated through the first neural network; Based on the latent variables specific to the aircraft wing shape sample, M fusion coefficients specific to the aircraft wing shape sample are determined through the first linear hypernetwork; The method further includes: Based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample, the parameter values ​​of the first linear supernetwork and the latent variables specific to each aircraft wing shape sample are also updated.

[0008] Optionally, each aircraft wing shape sample includes multiple spatial location points with a first resolution; the M grid basis function features of the aircraft wing shape sample include: M grid basis function features of the multiple spatial location points with the first resolution; the modulation features specific to the aircraft wing shape sample include: modulation features of the multiple spatial location points with the first resolution, wherein the modulation feature of each spatial location point is obtained by fusing the M grid basis function features of the spatial location point according to the M fusion coefficients specific to the aircraft wing shape sample. The aircraft wing shape sample is processed using the M grid basis functions at M resolutions to obtain the M grid basis function features of the aircraft wing shape sample, including: For a first spatial location point among multiple spatial location points of the first resolution, when the m-th grid basis function corresponds to the first resolution, locate multiple vertices of adjacent first grids from the grid of the first resolution; The m-th grid basis function is used to process the vertices of the plurality of adjacent first grids at the first resolution to obtain the grid basis function features of the plurality of adjacent first grid vertices. The grid basis function features of the plurality of adjacent first grid vertices are then interpolated to obtain the m-th grid basis function feature of the first spatial location point. When the (m+1)th grid basis function corresponds to a second resolution different from the first resolution, the first spatial location point is normalized, and the grid of the second resolution is normalized. Based on the normalized first spatial location points, locate multiple vertices of adjacent second grids from the normalized second resolution grid; By processing the vertices of the plurality of adjacent second grids at the second resolution using the (m+1)th grid basis function, the grid basis function features of the plurality of adjacent second grid vertices are obtained. Then, the grid basis function features of the plurality of adjacent second grid vertices are interpolated to obtain the (m+1)th grid basis function feature of the first spatial location point.

[0009] Optionally, the method further includes: For the first spatial location point The m-th grid basis function feature Reconstruct according to the following formula:

[0010] in, These are the characteristics of the basis functions. along coordinate axes coordinate axes One-dimensional basis vectors of the coordinate axes; Indicates the outer product; Store the basis function features along coordinate axes coordinate axes One-dimensional basis vectors of the coordinate axes.

[0011] Optionally, the aircraft wing shape sample set is the input physical field; when the first neural network is trained, the latent variables specific to each aircraft wing shape sample in the aircraft wing shape sample set are obtained; The flow field sample set corresponding to the aircraft wing shape sample set is the output physical field. One aircraft wing shape sample corresponds to one flow field sample to characterize the air velocity at each spatial location point of the aircraft wing shape sample. The method further includes: The flow field sample set is input into the second neural network to be trained; When the second neural network is trained, the latent variables specific to each flow field sample in the flow field sample set are obtained; Based on the latent variables specific to each aircraft wing shape sample in the aircraft wing shape sample set and the latent variables specific to each flow field sample in the flow field sample set, a processing network is trained. The trained processing network is used to establish a nonlinear mapping relationship between the latent variables of the input physical field and the latent variables of the output physical field.

[0012] Optionally, the method further includes: Input the wing shape of the target aircraft into the trained first neural network; The trained first neural network generates latent variables specific to the wing shape of the target aircraft. Based on the latent variables specific to the wing shape of the target aircraft, M fusion coefficients specific to the wing shape of the target aircraft are determined through the trained first linear supernetwork. The aircraft wing shape sample is processed at M resolutions using the M grid basis functions of the first trained neural network to obtain the M grid basis function features of the target aircraft wing shape. Based on the M fusion coefficients specific to the target aircraft wing shape, the M grid basis function features of the target aircraft wing shape are fused to obtain the modulation features specific to the target aircraft wing shape. The common features of the target aircraft wing shape and each aircraft wing shape sample in the aircraft wing shape sample set are captured by the first network weights of the first neural network after training. Based on the modulation features specific to the target aircraft wing shape sample and the common features of the target, the predicted features of the target aircraft wing shape sample are obtained; The wing shape of the target aircraft is reconstructed using the predicted features of the target aircraft's wing shape; While keeping the values ​​of the first network weights of the first neural network frozen after training, the latent variables specific to the target aircraft wing shape are updated with the goal of minimizing the difference between the reconstruction result of the target aircraft wing shape sample and the target aircraft wing shape. After updating the latent variables specific to the target aircraft wing shape, the trained processing network is used to convert the updated latent variables specific to the target aircraft wing shape into latent variables specific to the target flow field. Based on the latent variables specific to the target flow field, the target flow field is obtained through a trained second neural network, which represents the air velocity at each spatial location point of the target aircraft wing shape.

[0013] A second aspect of this application discloses a training device for a physical field neural network with a cross-scale separable grid representation, the device comprising: The sample input module is used to input the aircraft wing shape sample set into the first neural network to be trained. The resolution allocation module is used to assign M resolutions to the M grid basis functions of the first neural network, with one grid basis function corresponding to one resolution; the M resolutions are used to balance global feature learning and local feature learning on the aircraft wing shape sample set. The feature extraction module is used to capture the common features among the various aircraft wing shape samples in the aircraft wing shape sample set by using the first network weights of the first neural network. The coefficient determination module is used to determine M fusion coefficients specific to each aircraft wing shape sample through the first neural network. The sample processing module is used to process the aircraft wing shape sample at M resolutions using the M grid basis functions to obtain the M grid basis function features of the aircraft wing shape sample; The feature fusion module is used to fuse the M grid basis function features of the aircraft wing shape sample according to the M fusion coefficients specific to the aircraft wing shape sample, so as to obtain the modulation features specific to the aircraft wing shape sample. The feature prediction module is used to obtain the predicted features of the aircraft wing shape sample based on the modulation features specific to the aircraft wing shape sample and the common features. The sample reconstruction module is used to reconstruct the wing shape sample of each aircraft using the predicted features of the sample. The parameter update module is used to update the values ​​of the first network weights of the first neural network and the parameter values ​​of the M grid basis functions based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample.

[0014] A third aspect of this application discloses an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the physical field neural network training method for cross-scale separable grid representation described in the first aspect of this application.

[0015] A fourth aspect of this application discloses a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the physical field neural network training method for cross-scale separable grid representation described in the first aspect of this application.

[0016] A fifth aspect of this application discloses a computer program product, including a computer program that, when executed by a processor, implements the steps of the physical field neural network training method for cross-scale separable grid representation described in the first aspect of this application.

[0017] The embodiments of this application have the following advantages: In this embodiment, a multi-scale spatial representation system is constructed by assigning M different resolutions to M grid basis functions. This enables low-resolution grid basis functions to capture the macroscopic contours and global topological structure of the aircraft wing shape, while high-resolution grid basis functions can characterize the fine geometric features of local areas, thereby improving the model's ability to express complex wing geometry.

[0018] For each aircraft wing shape sample, a first neural network determines M fusion coefficients specific to that sample. Based on these coefficients, the multi-resolution grid basis function features are weighted and fused to obtain sample-specific modulation features. This mechanism enables the model to generate personalized spatial modulation parameters for each specific aircraft wing shape instance, while sharing the weights of the first network to capture common features among samples. This enhances the model's adaptive representation capability for samples of different shapes and allows for more accurate reconstruction of complex and varied wing geometries.

[0019] Furthermore, by jointly updating the parameters of the first network weights and M grid basis functions end-to-end based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample, it is ensured that the multi-resolution grid basis functions, the fusion coefficient generation module, and the shared feature extraction network can be optimized collaboratively. This enables the grid basis functions to learn spatial bases that are adapted to specific physical field modeling tasks, and the shared weights to extract common patterns among samples, thereby improving the overall model representation efficiency and reconstruction accuracy. Attached Figure Description

[0020] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the description of the embodiments of this application will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a flowchart illustrating the steps of a physical field neural network training method with cross-scale separable grid representation provided in an embodiment of this application. Figure 2 This is a schematic diagram illustrating the positioning of adjacent grid vertices under different resolution grids, provided in an embodiment of this application. Figure 3 This is a flowchart illustrating the steps of a flow field prediction method for aircraft wing shape provided in an embodiment of this application. Figure 4 This is a schematic diagram of the structure of a physical field neural network training device with a cross-scale separable grid representation provided in an embodiment of this application; Figure 5 This is a schematic diagram of the structure of an electronic device provided in an embodiment of this application. Detailed Implementation

[0022] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0023] Reference Figure 1 As shown, Figure 1 This is a flowchart illustrating the steps of a physical field neural network training method based on a cross-scale separable grid representation provided in an embodiment of this application. Figure 1 As shown, the method may include steps S110 to S190: Step S110: Input the aircraft wing shape sample set into the first neural network to be trained.

[0024] In this step, the aircraft wing shape sample set contains multiple aircraft wing shape samples, each describing the wing geometry of a specific configuration. In practice, each sample can be represented as a function defined in the spatial domain, for example, discretized using geometric parameters (such as surface height, thickness distribution, etc.) at spatial coordinate points. These samples are then used as input data to the first neural network to be trained, for subsequent feature extraction and model optimization.

[0025] Step S120: Assign M resolutions to the M grid basis functions of the first neural network, with one grid basis function corresponding to one resolution; the M resolutions are used to balance global feature learning and local feature learning on the aircraft wing shape sample set.

[0026] In this step, a multi-resolution spatial representation system is constructed for the first neural network. Specifically, M grid basis functions are introduced into the first neural network, and each grid basis function is assigned an independent spatial resolution. These resolutions are arranged from low to high, with lower-resolution grid basis functions covering a larger spatial range, aiming to capture global features such as the macroscopic contour of the aircraft wing shape and large-scale changes in the wingspan direction; higher-resolution grid basis functions focus on local regions, used to characterize subtle curvature changes and local geometric features at locations such as the wingtip and leading edge. This multi-resolution design allows the model to model spatial information at different scales separately, avoiding the mixing of high-frequency and low-frequency components in a single scale.

[0027] Step S130: For the aircraft wing shape sample set, capture the common features among the various aircraft wing shape samples in the aircraft wing shape sample set through the first network weights of the first neural network.

[0028] In this step, the first neural network contains a set of shared network weights (i.e., the first network weights). These first network weights are jointly optimized across all aircraft wing shape samples during training to learn common features (common patterns) among different samples. For example, different wing shapes all follow basic aerodynamic constraints in their spatial structure, sharing commonalities in geometric continuity and smoothness. By extracting these common structural priors from a large number of samples, the first network weights form a general representation capability for aircraft wing shapes.

[0029] Step S140: For each aircraft wing shape sample, determine M fusion coefficients specific to that aircraft wing shape sample using the first neural network.

[0030] In this step, a set of dedicated fusion coefficients is generated for each aircraft wing shape sample. These fusion coefficients are used for subsequent weighted fusion of basis function features across different resolution grids. The fusion coefficients can be generated in various ways, such as mapping them from the latent variables corresponding to the samples via a hypernetwork, or by directly learning the coefficient vector corresponding to each sample. This step enables the model to adaptively adjust the contribution of features at different scales to the final representation based on the unique geometric characteristics of each aircraft wing shape sample.

[0031] Step S150: Process the aircraft wing shape sample at M resolutions using the M grid basis functions to obtain the M grid basis function features of the aircraft wing shape sample.

[0032] In this step, for a given aircraft wing shape sample, spatial features are extracted from the sample using the M grid basis functions with different resolutions constructed in step S120. Specifically, for each spatial location point, the adjacent vertices of that point in each resolution grid are located, and the grid basis function values ​​at the vertices are interpolated (e.g., bilinear or trilinear interpolation) to obtain the grid basis function features of that location point at each resolution. After this step, each sample obtains M spatial features at each spatial location point.

[0033] Step S160: Based on the M fusion coefficients specific to the aircraft wing shape sample, fuse the M grid basis function features of the aircraft wing shape sample to obtain the modulation features specific to the aircraft wing shape sample.

[0034] In this step, the M fusion coefficients obtained in step S140 are used to perform weighted fusion of the M grid basis function features obtained in step S150. The fusion method can be linear weighted summation or other fusion strategies. Through this operation, multi-scale spatial features are combined according to the individual needs of the sample to generate modulation features specific to the current sample. These modulation features reflect the comprehensive geometric information of the sample at different spatial scales and have instance-specific characteristics.

[0035] Step S170: Based on the modulation features specific to the aircraft wing shape sample and the common features, obtain the predicted features of the aircraft wing shape sample.

[0036] In this step, the specific modulation features of the aircraft wing shape sample obtained in step S160 are combined with the common features extracted by the first network weights in step S130 to jointly generate the predicted features of the sample. In a specific implementation, this combination can be reflected in the adjustment of the activation values ​​of the hidden layer of the neural network by the modulation features. For example, the modulation features can be superimposed as a bias term on the output of the network layer to achieve instantiation modulation of the shared features.

[0037] As an example implementation, SIREN (Sinusoidal Representation Networks) can be used as the backbone architecture, and the activation function of its hidden layer can be expressed as:

[0038] in, Denotes the activation function of the i-th hidden layer; The input activation value of the i-th hidden layer (i.e., the output of the previous layer); x is the spatial coordinate point of the input; It is a sinusoidal activation function; This is the frequency scaling factor, used to control the frequency of the sine function; and These are the weight matrix and bias term of the i-th hidden layer, respectively; is the modulation feature of the i-th hidden layer, which is an introduced spatial correlation modulation term that acts on the network's bias term to achieve instantiation modulation of the shared features.

[0039] It can be represented as:

[0040] in, The fusion coefficients are the basis functions of the m-th grid corresponding to the i-th hidden layer. The value of the m-th grid basis function at spatial location x is obtained through multi-resolution grid interpolation.

[0041] After this step, the predicted features output by the model include both general structural priors learned from a large number of samples and geometric details unique to the current sample.

[0042] Step S180: Reconstruct the aircraft wing shape sample using the predicted features of each aircraft wing shape sample.

[0043] In this step, based on the predicted features obtained in step S170, the features are mapped back to the original physical space through the output layer of the neural network to generate the reconstructed aircraft wing shape. The reconstruction process aims to make the neural network output approximate the original input sample as closely as possible, thereby verifying the model's ability to learn and represent sample features.

[0044] Step S190: Based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample, update the values ​​of the first network weights of the first neural network and the parameter values ​​of the M grid basis functions.

[0045] In this step, a loss function is constructed to measure the difference between the reconstructed wing shape sample and the original wing shape sample. For example, mean squared error (MSE) can be used as the loss function to calculate the sum of squares of the differences between the predicted and true values ​​at all spatial locations. Then, using the backpropagation algorithm, the first network weights of the first neural network and the parameter values ​​of the M grid basis functions are jointly optimized and updated based on this loss function. During training, the shared weights and multi-resolution grid basis functions are adjusted collaboratively, allowing the model to gradually learn the optimal spatial representation.

[0046] The technical solution adopted in this embodiment constructs a multi-resolution grid basis function system, enabling the model to separately handle the global trend and local high-frequency details of the physical field, achieving decoupled modeling at the spatial scale. Low-resolution grid basis functions capture the macroscopic contour and global topological structure of the aircraft wing shape, while high-resolution grid basis functions characterize the fine geometric features of local regions, thereby improving the model's ability to express complex wing geometries. Simultaneously, by generating dedicated fusion coefficients for each sample and weighted fusion of the multi-resolution grid basis function features, instance-adaptive spatial modulation is achieved. Based on the extraction of common features among samples using shared network weights, each sample can obtain personalized modulation features, enhancing the model's ability to accurately reconstruct wing shapes with different configurations. Furthermore, by incorporating the parameter values ​​of the first network weights and the multi-resolution grid basis functions into a unified training framework, collaborative optimization is achieved through end-to-end joint updates. This allows the grid basis functions to learn a spatial basis adapted to a specific physical field modeling task, and the shared weights can extract common patterns among samples, thereby improving the overall model's representation efficiency and reconstruction accuracy.

[0047] In an optional embodiment, step S140 above, "for each aircraft wing shape sample, determine M fusion coefficients specific to that aircraft wing shape sample using the first neural network," can be generated using a combination of latent variables and a hypernetwork. Specifically, it can include the following steps S140-1 to S140-2: Step S140-1: For each aircraft wing shape sample, generate latent variables specific to that aircraft wing shape sample using the first neural network.

[0048] In this step, a unique latent variable is assigned to each aircraft wing shape sample to characterize the instance-specific information of that sample. The latent variable can be viewed as a compact representation of the sample in the latent space, with a dimension lower than that of the original physical field, enabling efficient encoding of the sample's personalized geometric features. In the initial training phase, the latent variable for each sample can be obtained through random initialization or a certain encoding method, and participates in updating as an optimizable parameter during the training process.

[0049] Step S140-2: Based on the latent variables specific to the aircraft wing shape sample, determine the M fusion coefficients specific to the aircraft wing shape sample through the first linear supernetwork; In this step, the latent variables obtained in step S140-1 are input into the first linear supernetwork, which is used to map the latent variables into M fusion coefficients. Specifically, the first linear supernetwork can be implemented in the form of linear mapping, that is, by using a linear transformation layer to convert the latent variables into a coefficient vector of dimension M.

[0050] For example, the process of generating M fusion coefficients specifically for aircraft wing shape samples can be represented as:

[0051] Where z represents a latent variable specific to the aircraft wing shape sample. This represents the first linear supernetwork. Here is the weight matrix of the first linear supernetwork. This is the bias term for the first linear supernetwork. These are the M fusion coefficients generated.

[0052] In this way, the latent variables of each aircraft wing shape sample are linearly transformed to obtain a set of fusion coefficients specific to that sample. These coefficients reflect the importance of the basis functions of each resolution grid in the current sample representation, thus achieving adaptive weighting of multi-scale features.

[0053] Furthermore, in addition to the steps described above, the method also includes the following step S200: Step S200: Based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample, the parameter values ​​of the first linear supernetwork and the latent variables specific to each aircraft wing shape sample are also updated.

[0054] In this step, the parameter values ​​of the first linear supernetwork (i.e., the weight matrix and bias terms of the supernetwork) and the latent variables specific to each sample are incorporated into a unified training framework. Specifically, based on the loss function described in step S190, the parameters of the first linear supernetwork and the latent variables of all samples are simultaneously optimized and updated using the backpropagation algorithm.

[0055] This joint update mechanism enables latent variables to learn the optimal representation of samples in the latent space, i.e., the latent variables gradually converge to an encoding that can accurately reconstruct the geometry of the corresponding samples; at the same time, the hypernetwork parameters are optimized, making the mapping relationship from latent variables to fusion coefficients more accurate. Through this end-to-end collaborative training, the model can automatically learn the coupling mode between "shared priors" (provided by the first network weights and grid basis functions) and "instance characteristics" (provided by latent variables and the hypernetwork).

[0056] The technical solution adopted in this embodiment represents the individual characteristics of samples through low-dimensional latent variables, compressing the geometric information of each aircraft wing shape sample into a low-dimensional space, thereby reducing the storage overhead of instance-related parameters. A first linear hypernetwork is used to map latent variables to fusion coefficients, realizing a linear transformation from the latent space to the multi-scale grid basis function weight space. Incorporating latent variables and hypernetwork parameters into an end-to-end joint update framework enables the model to simultaneously optimize the representation quality of the latent space and the generation accuracy of the fusion coefficients, further improving the model's ability to reconstruct complex wing geometries and its training convergence efficiency.

[0057] In one optional embodiment, each aircraft wing shape sample includes multiple spatial location points with a first resolution; the M grid basis function features of the aircraft wing shape sample include: M grid basis function features of the multiple spatial location points with the first resolution; the modulation features specific to the aircraft wing shape sample include: modulation features of the multiple spatial location points with the first resolution, wherein the modulation feature of each spatial location point is obtained by fusing the M grid basis function features of the spatial location point according to the M fusion coefficients specific to the aircraft wing shape sample.

[0058] Furthermore, step S150 above, "processing the aircraft wing shape sample at M resolutions using the M grid basis functions to obtain the M grid basis function features of the aircraft wing shape sample," may include the following steps S150-1 to S150-5: Step S150-1: For the first spatial location point among multiple spatial location points of the first resolution, and with the m-th grid basis function corresponding to the first resolution, locate the vertices of multiple adjacent first grids from the grid of the first resolution.

[0059] In this step, for the currently processed spatial location point (referred to as the first spatial location point), when the resolution of the m-th grid basis function is the same as the original resolution of the sample to which the location point belongs, the adjacent grid vertices around the location point are located in the grid structure of that resolution. For example, in two-dimensional space, for a location point with coordinates (x, y), four adjacent grid vertices around it are located; in three-dimensional space, eight adjacent grid vertices around it are located.

[0060] Step S150-2: Process the vertices of the plurality of adjacent first grids at the first resolution using the m-th grid basis function to obtain the grid basis function features of the plurality of adjacent first grid vertices, and perform interpolation calculation on the grid basis function features of the plurality of adjacent first grid vertices to obtain the m-th grid basis function feature of the first spatial location point.

[0061] In this step, the grid basis function values ​​at each adjacent grid vertex are obtained. These values ​​are predefined parameters of the grid basis function at the vertex positions for that resolution. Then, the characteristic values ​​of the grid basis function at the first spatial location point are obtained by interpolating these vertex values. The interpolation method can be selected according to the spatial dimension; for example, bilinear interpolation is used in two dimensions, and trilinear interpolation is used in three dimensions.

[0062] Step S150-3: When the (m+1)th grid basis function corresponds to a second resolution different from the first resolution, normalize the first spatial location point and normalize the grid of the second resolution.

[0063] In this step, when processing basis functions for grids with different original resolutions than the current spatial location point, coordinate normalization is required because the spatial coordinate ranges of grids with different resolutions may differ. Specifically, the original coordinates of the first spatial location point are mapped to a normalized coordinate space, and the coordinate range of the second resolution grid is also normalized accordingly, ensuring that both are aligned in the same reference frame. This normalization process ensures the correctness and consistency of cross-resolution interpolation.

[0064] Step S150-4: Based on the normalized first spatial location points, locate the vertices of multiple adjacent second grids from the normalized second resolution grid.

[0065] In this step, the normalized coordinates are used to locate the adjacent grid vertices around the given point in the normalized grid at the second resolution. For example... Figure 2 As shown, the aircraft wing shape sample includes For multiple spatial locations, the spatial location (1,1) can be normalized to... For a resolution of The grid can locate the vertices of four adjacent second grid cells: A, B, C, and D; for a resolution of... The grid can locate the vertices of four adjacent second grid cells: E, E', G, and H. Step S150-5: Process the vertices of the plurality of adjacent second grids at the second resolution using the (m+1)th grid basis function to obtain the grid basis function features of the plurality of adjacent second grid vertices, and perform interpolation calculation on the grid basis function features of the plurality of adjacent second grid vertices to obtain the (m+1)th grid basis function feature of the first spatial location point.

[0066] In this step, the grid basis function values ​​at each adjacent vertex in the second-resolution grid are obtained, and the grid basis function characteristic values ​​of the first spatial location point at the second resolution are calculated by interpolation. This interpolation calculation is similar to step S150-2, but is applied to the grid vertex values ​​at the second resolution.

[0067] By cyclically executing steps S150-1 to S150-5, for each spatial location point, the above processing is performed on M grid basis functions of different resolutions to obtain the grid basis function characteristics of that location point at M resolutions.

[0068] The technical solution adopted in this embodiment uses grid vertex interpolation to enable grid basis functions to provide smooth spatial feature representations for arbitrary coordinate positions within a continuous spatial domain. Compared to directly using discrete grid points, the interpolation mechanism ensures the spatial continuity of features, allowing the neural network to perform inference at any resolution. Through the collaborative processing of multi-resolution grids, each spatial location point can obtain spatial feature representations at multiple scales. Low-resolution grid basis functions cover the global range with a larger grid spacing, providing macroscopic structural information; high-resolution grid basis functions characterize local details with fine grids, and together they constitute a complete multi-scale spatial representation. Through coordinate normalization processing, the problem of inconsistency in coordinate systems between grids of different resolutions is solved, ensuring the accuracy and consistency of cross-scale feature extraction, and enabling multi-resolution grid basis functions to work collaboratively in the same spatial reference frame.

[0069] In an optional embodiment, the method further includes steps A1 to A2: Step A1: For the first spatial location point The m-th grid basis function feature Reconstruct according to the following formula:

[0070] in, These are the characteristics of the basis functions. along coordinate axes coordinate axes One-dimensional basis vectors of the coordinate axes; It represents the outer product.

[0071] Step A2: Store the basis function features along coordinate axes coordinate axes One-dimensional basis vectors of the coordinate axes.

[0072] In this embodiment, for high-dimensional physics field modeling tasks (such as three-dimensional aircraft wing flow field analysis), the traditional complete mesh tensor storage method results in a cubic increase in the number of parameters with increasing spatial resolution. For example, for a resolution of... A three-dimensional mesh, the complete tensor needs to be stored There are 2.09 million parameters. When N is 128, the number of parameters reaches approximately 2.09 million, which will bring huge storage overhead and computational burden in practical applications.

[0073] To address the aforementioned issues, this embodiment introduces a spatially separable tensor decomposition technique. Specifically, for the m-th grid basis function at its three-dimensional location... value at Instead of storing it as a complete three-dimensional tensor, it is decomposed into the outer product of three independent one-dimensional basis vectors.

[0074] Through the above decomposition method, the original storage required A complete 3D mesh tensor with 3N parameters is compressed to require storing only three 1D vectors, for a total of 3N parameters. Taking N=128 as an example, approximately 2.09 million parameters needed to be stored before compression, while only 384 parameters need to be stored after compression, a compression ratio of approximately 5440 times. The compression effect is even more significant for higher resolution meshes.

[0075] The technical solution adopted in this embodiment decomposes the complete high-dimensional grid tensor into a combination of multiple one-dimensional basis vectors through outer products. This reduces the number of parameters, which originally increased exponentially with spatial resolution, to a linear increase, thereby reducing the model's storage overhead and GPU memory usage. During the forward propagation of the neural network, accessing the grid basis function values ​​through the decomposed storage method only requires index reading and multiplication operations of one-dimensional vectors, avoiding the complete storage and access of the high-dimensional tensor and improving computational efficiency. The forced separability of the grid basis functions along each spatial axis aligns with the variable separation characteristics present in many physical field problems, helping the model learn spatial representations with greater physical interpretability and generalization ability, reducing parameter redundancy, and improving the model's parameter utilization efficiency.

[0076] In one optional embodiment, the aircraft wing shape sample set is the input physical field; when the first neural network is trained, the latent variable specific to each aircraft wing shape sample in the aircraft wing shape sample set is obtained; the flow field sample set corresponding to the aircraft wing shape sample set is the output physical field, and one aircraft wing shape sample corresponds to one flow field sample to characterize the air velocity at each spatial location point of the aircraft wing shape sample.

[0077] Specifically, the aircraft wing shape sample set serves as the input physical field; that is, the aircraft wing shape samples used for training constitute the input function space, used to describe the geometric boundary conditions or initial configuration of the physical system. Upon completion of the first neural network training, a latent variable specific to each aircraft wing shape sample in the aircraft wing shape sample set is obtained. As previously mentioned, this latent variable is a low-dimensional vector generated in step S140-1 and jointly optimized and updated in step S200; it compactly encodes the personalized geometric features of each wing shape sample.

[0078] Flow field samples are typically obtained through numerical simulations (such as computational fluid dynamics (CFD) simulations) or experimental measurements, describing the airflow velocity distribution (including velocity magnitude and direction) at various points in the spatial domain under specific airfoil shape conditions. A complex nonlinear mapping relationship exists between the input and output physical fields, determined by the fluid dynamics governing equations.

[0079] Furthermore, in addition to the steps described above, the method also includes the following steps B1 to B3: Step B1: Input the flow field sample set into the second neural network to be trained.

[0080] In this step, a second neural network is constructed, whose network architecture can be the same as or similar to that of the first neural network. The flow field sample set is used as input data and provided to the second neural network to be trained. This network is used to learn the spatial representation of the output physical field, and its structure is similar to that of the first neural network, including shared network weights, multi-resolution grid basis functions, and a latent variable generation mechanism.

[0081] Step B2: When the second neural network has been trained, obtain the latent variables specific to each flow field sample in the flow field sample set.

[0082] In this step, the second neural network is trained using a similar training process to the first neural network. Specifically, by assigning multi-resolution to the grid basis functions of the second neural network, shared weights are used to capture common features among the flow field samples. A unique latent variable is generated for each flow field sample, and the network parameters are updated based on the difference between the reconstructed result and the original flow field sample. After training, each flow field sample corresponds to a latent variable, which compactly encodes the spatial distribution characteristics of that flow field sample.

[0083] Step B3: Based on the latent variables specific to each aircraft wing shape sample in the aircraft wing shape sample set and the latent variables specific to each flow field sample in the flow field sample set, train the processing network. The trained processing network is used to establish a nonlinear mapping relationship between the latent variables of the input physical field and the latent variables of the output physical field.

[0084] In this step, the input latent variables obtained from training the first neural network are... (Corresponding to the wing shape) and the output latent variables obtained from the training of the second neural network (Corresponding to the flow field) forms paired data, which is used to train a processing network. This processing network typically employs a multilayer perceptron (MLP) or other network structures capable of fitting nonlinear functions. Its input consists of the latent variables of the wing shape sample, and its output consists of the latent variables of the corresponding flow field sample. By minimizing the difference between the predicted and true latent variables (e.g., mean squared error), the processing network is trained to learn a nonlinear mapping from the input latent space to the output latent space, i.e.: .

[0085] It should be noted that the specific architecture of the processing network can be designed according to the complexity of the task. For relatively simple mapping relationships, shallow neural networks can be used; for complex nonlinear mappings (such as turbulence evolution, cross-condition prediction, etc.), deep networks or structures such as residual connections can be used. This application does not impose any restrictions on this.

[0086] Through steps B1 to B3 above, a two-stage training strategy is constructed. In the first stage, two independent neural networks (the first neural network and the second neural network) are used to perform autoencoder training on the input and output physical fields respectively to obtain their respective latent space representations. In the second stage, a processing network is used to learn the mapping relationship between the two latent spaces. This staged training strategy decomposes the complex high-dimensional function space mapping problem into a mapping problem between two low-dimensional latent spaces, reducing the learning difficulty.

[0087] The technical solution adopted in this embodiment uses two neural networks to autoencode and train the input and output physical fields respectively, mapping high-dimensional continuous spatial functions into low-dimensional latent variables. This reduces the data dimensionality and provides a compact and efficient feature representation for subsequent mapping learning. The complex operator learning task is decomposed into two independent stages: the first stage focuses on learning the spatial structure features of the physical field itself, and the second stage focuses on learning the mapping relationships in the latent space (the mapping from input latent variables to output latent variables). This decoupled design makes the task of each stage clearer and reduces the complexity of model training.

[0088] In one alternative embodiment, such as Figure 3 As shown, Figure 3 This is a flowchart illustrating the steps of a flow field prediction method for aircraft wing shape provided in an embodiment of this application. Specifically, the method includes steps C1 to C11: Step C1: Input the target aircraft wing shape into the trained first neural network.

[0089] In this step, the wing shape of the target aircraft to be predicted is obtained. This shape is a new, untrained wing geometry. This wing shape is then input into the first, already trained neural network as input for the encoding process.

[0090] Step C2: Generate latent variables specific to the wing shape of the target aircraft using the first trained neural network.

[0091] In this step, a latent variable is initialized for the wing shape of the target aircraft. This latent variable can be initialized randomly or heuristically based on its similarity to known samples. This latent variable will be adjusted through optimization in subsequent steps.

[0092] Step C3: Based on the latent variables specific to the wing shape of the target aircraft, determine the M fusion coefficients specific to the wing shape of the target aircraft through the trained first linear supernetwork.

[0093] In this step, the latent variables from step C2 are input into the trained first linear supernetwork, and M fusion coefficients are generated through the linear mapping of the supernetwork. This process is similar to step S140-2 in the training phase, but uses the fixed parameters of the trained supernetwork.

[0094] Step C4: Process the aircraft wing shape sample at M resolutions using the M grid basis functions of the trained first neural network to obtain the M grid basis function features of the target aircraft wing shape.

[0095] In this step, the M trained grid basis functions are used to process the spatial location points of the target aircraft wing shape according to the cross-resolution interpolation method described in the previous embodiment, so as to obtain the M grid basis function features.

[0096] Step C5: Based on the M fusion coefficients specific to the target aircraft wing shape, fuse the M grid basis function features of the target aircraft wing shape to obtain the modulation features specific to the target aircraft wing shape.

[0097] In this step, the M fusion coefficients obtained in step C3 are weighted and fused with the M grid basis function features obtained in step C4 to generate a modulation feature specific to the shape of the target aircraft wing. This modulation feature is used for subsequent spatial modulation of the activation states of the hidden layer of the neural network.

[0098] Step C6: Using the first network weights of the trained first neural network, capture the common features between the target aircraft wing shape and each aircraft wing shape sample in the aircraft wing shape sample set.

[0099] In this step, the weights of the trained first network are used to process the target aircraft wing shape, extracting the structural features common to all samples in the training sample set. These common features reflect the general prior structure of the aircraft wing shape.

[0100] Step C7: Based on the modulation features specific to the target aircraft wing shape sample and the common features of the target, obtain the predicted features of the target aircraft wing shape sample.

[0101] In this step, the modulation features obtained in step C5 are combined with the common features obtained in step C6 to generate predicted features of the target aircraft wing shape. In a specific implementation, this combination can be reflected in the modulation features adjusting the activation values ​​of the hidden layers of the neural network.

[0102] Step C8: Reconstruct the wing shape of the target aircraft using the predicted features of the wing shape of the target aircraft.

[0103] In this step, based on the predicted features obtained in step C7, the shape of the target aircraft wing is reconstructed through the output layer of the neural network to obtain the reconstructed geometry.

[0104] Step C9: While keeping the values ​​of the first network weights of the trained first neural network frozen, update the latent variables specific to the target aircraft wing shape with the goal of minimizing the difference between the reconstruction result of the target aircraft wing shape sample and the target aircraft wing shape.

[0105] In this step, the difference between the reconstructed result and the original target aircraft wing shape is calculated as the loss. Unlike the training phase, this step keeps the first network weights (shared weights) and the parameter values ​​of the grid basis functions of the first neural network frozen, and only optimizes and updates the latent variables of the target aircraft wing shape. The values ​​of the latent variables are iteratively adjusted using the backpropagation algorithm until the reconstruction loss converges. The purpose of this process is to map the target aircraft wing shape to the pre-trained latent space to obtain the optimal latent variable representation.

[0106] For example, the loss function corresponding to this optimization process It can be represented as:

[0107] in, This is a spatial domain, typically a bounded physical region, used to define the sampling range of coordinate point x; For the first parameterized neural network, This represents the shared network weight (first network weight). Indicates by latent variables Modulation parameters generated by a hypernetwork; Let i target aircraft wing shape samples be used as the input physical field; The mathematical expectation symbol represents the expectation of a spatial domain. The average of x is taken from all sampling points within the range; This represents the reconstructed output value of the first neural network at spatial location point x. The value of the target aircraft wing shape sample at spatial location point x.

[0108] Step C10: After updating the latent variables specific to the target aircraft wing shape, the trained processing network is used to convert the updated latent variables specific to the target aircraft wing shape into latent variables specific to the target flow field.

[0109] In this step, the optimal latent variables of the target aircraft wing shape obtained in step C9 are input into the trained processing network. This processing network was trained in step B3, establishing a nonlinear mapping relationship between the input physical field latent variables and the output physical field latent variables. After processing the network forward propagation, the corresponding latent variables specific to the target flow field are output.

[0110] Step C11: Based on the latent variables specific to the target flow field, the target flow field is obtained through the trained second neural network. The target flow field represents the air velocity at each spatial location point of the target aircraft wing shape.

[0111] In this step, the latent variables of the target flow field obtained in step C10 are input into the trained second neural network. The second neural network uses its shared weights, multi-resolution grid basis functions, and supernetwork to decode the latent variables into a complete flow field distribution, namely the air velocity field corresponding to the shape of the target aircraft wing. This flow field contains air velocity information (including velocity magnitude and direction) at each spatial location point.

[0112] Based on the above steps, this embodiment achieves flow field prediction for a new input aircraft wing shape through a complete inference process. For a new target aircraft wing shape, only a small number of latent variable parameters need to be optimized during testing, and the corresponding flow field can be quickly mapped by the processing network. Furthermore, by freezing the weights of the first network and only optimizing the latent variables for encoding, the model can adapt to wing geometries of arbitrary shapes without retraining the entire network. The processing network establishes mapping relationships in the latent space, which has stronger generalization ability compared to directly learning mappings in a high-dimensional function space. The entire inference process follows a clear logic of "encoding-latent variable mapping-decoding," with clear division of labor at each stage: the first neural network is responsible for the compact representation of the input field, the processing network is responsible for learning the mapping in the latent space, and the second neural network is responsible for the accurate reconstruction of the output field.

[0113] This application also provides a training device for a physical field neural network with a cross-scale separable grid representation, referring to... Figure 4 As shown, Figure 4 This is a schematic diagram of a physical field neural network training device with a cross-scale separable grid representation provided in an embodiment of this application. The device includes: The sample input module 410 is used to input the aircraft wing shape sample set into the first neural network to be trained. The resolution allocation module 420 is used to allocate M resolutions to the M grid basis functions of the first neural network, with one grid basis function corresponding to one resolution; the M resolutions are used to take into account both global feature learning and local feature learning of the aircraft wing shape sample set. Feature extraction module 430 is used to capture the common features among the various aircraft wing shape samples in the aircraft wing shape sample set by using the first network weights of the first neural network. The coefficient determination module 440 is used to determine M fusion coefficients specific to each aircraft wing shape sample through the first neural network. The sample processing module 450 is used to process the aircraft wing shape sample at M resolutions using the M grid basis functions to obtain the M grid basis function features of the aircraft wing shape sample. The feature fusion module 460 is used to fuse the M grid basis function features of the aircraft wing shape sample according to the M fusion coefficients specific to the aircraft wing shape sample, so as to obtain the modulation features specific to the aircraft wing shape sample. The feature prediction module 470 is used to obtain the predicted features of the aircraft wing shape sample based on the modulation features specific to the aircraft wing shape sample and the common features. The sample reconstruction module 480 is used to reconstruct the aircraft wing shape sample using the predicted features of each aircraft wing shape sample; The parameter update module 490 is used to update the values ​​of the first network weights of the first neural network and the parameter values ​​of the M grid basis functions based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample.

[0114] It is understood that the physical field neural network training device with cross-scale separable grid representation in the embodiments of this application can implement the physical field neural network training method with cross-scale separable grid representation in the above embodiments. The physical field neural network training device with cross-scale separable grid representation has the same advantages as the physical field neural network training method with cross-scale separable grid representation in the above embodiments compared with the prior art, and will not be repeated here.

[0115] This application also provides an electronic device, see embodiments thereof. Figure 5 , Figure 5 This is a schematic diagram of the structure of an electronic device provided in an embodiment of this application. For example... Figure 5 As shown, the electronic device 500 includes a memory 510 and a processor 520. The memory 510 and the processor 520 are connected via a bus for communication. The memory 510 stores a computer program that can run on the processor 520 to implement the steps of the physical field neural network training method for cross-scale separable grid representation described in the embodiments of this application.

[0116] This application also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the physical field neural network training method for cross-scale separable grid representation described in this application.

[0117] This application also provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the physical field neural network training method for cross-scale separable grid representation described in this application.

[0118] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0119] This application describes embodiments of methods and apparatus according to flowchart illustrations and / or block diagrams. It should be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing terminal device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing terminal device, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0120] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing terminal device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0121] These computer program instructions can also be loaded onto a computer or other programmable data processing terminal equipment, causing a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0122] Although preferred embodiments of the present application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the embodiments of the present application.

[0123] Finally, it should be noted that in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or terminal device that includes said element.

[0124] The foregoing has provided a detailed description of a training method, apparatus, device, medium, and program product for a physical field neural network with a cross-scale separable grid representation. Specific examples have been used to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the method and its core ideas. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this application. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A method for training a physical field neural network with a cross-scale separable grid representation, characterized in that, include: Input the set of aircraft wing shape samples into the first neural network to be trained; M resolutions are assigned to the M grid basis functions of the first neural network, with one grid basis function corresponding to one resolution; the M resolutions are used to balance global feature learning and local feature learning on the aircraft wing shape sample set; For the aircraft wing shape sample set, the common features among the various aircraft wing shape samples in the aircraft wing shape sample set are captured through the first network weights of the first neural network; For each aircraft wing shape sample, the first neural network determines M fusion coefficients specific to that aircraft wing shape sample. The aircraft wing shape sample is processed using the M grid basis functions at M different resolutions to obtain the M grid basis function features of the aircraft wing shape sample. Based on the M fusion coefficients specific to the aircraft wing shape sample, the M grid basis function features of the aircraft wing shape sample are fused to obtain the modulation features specific to the aircraft wing shape sample. Based on the modulation features specific to the aircraft wing shape sample and the common features, the predicted features of the aircraft wing shape sample are obtained; The predicted features of each aircraft wing shape sample are used to reconstruct the aircraft wing shape sample; Based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample, the values ​​of the first network weights of the first neural network and the parameter values ​​of the M grid basis functions are updated.

2. The method according to claim 1, characterized in that, For each aircraft wing shape sample, the first neural network determines M fusion coefficients specific to that aircraft wing shape sample, including: For each aircraft wing shape sample, latent variables specific to that aircraft wing shape sample are generated through the first neural network; Based on the latent variables specific to the aircraft wing shape sample, M fusion coefficients specific to the aircraft wing shape sample are determined through the first linear hypernetwork; The method further includes: Based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample, the parameter values ​​of the first linear supernetwork and the latent variables specific to each aircraft wing shape sample are also updated.

3. The method according to claim 1, characterized in that, Each aircraft wing shape sample includes multiple spatial location points with a first resolution; The M grid basis function features of the aircraft wing shape sample include: M grid basis function features of multiple spatial location points at a first resolution; the modulation features specific to the aircraft wing shape sample include: modulation features of multiple spatial location points at a first resolution, wherein the modulation feature of each spatial location point is obtained by fusing the M grid basis function features of that spatial location point with the M fusion coefficients specific to the aircraft wing shape sample. The aircraft wing shape sample is processed using the M grid basis functions at M resolutions to obtain the M grid basis function features of the aircraft wing shape sample, including: For a first spatial location point among multiple spatial location points of the first resolution, when the m-th grid basis function corresponds to the first resolution, locate multiple vertices of adjacent first grids from the grid of the first resolution; The m-th grid basis function is used to process the vertices of the plurality of adjacent first grids at the first resolution to obtain the grid basis function features of the plurality of adjacent first grid vertices. The grid basis function features of the plurality of adjacent first grid vertices are then interpolated to obtain the m-th grid basis function feature of the first spatial location point. When the (m+1)th grid basis function corresponds to a second resolution different from the first resolution, the first spatial location point is normalized, and the grid of the second resolution is normalized. Based on the normalized first spatial location points, locate multiple vertices of adjacent second grids from the normalized second resolution grid; By processing the vertices of the plurality of adjacent second grids at the second resolution using the (m+1)th grid basis function, the grid basis function features of the plurality of adjacent second grid vertices are obtained. Then, the grid basis function features of the plurality of adjacent second grid vertices are interpolated to obtain the (m+1)th grid basis function feature of the first spatial location point.

4. The method according to claim 3, characterized in that, The method further includes: For the first spatial location point The m-th grid basis function feature Reconstruct according to the following formula: , in, These are the characteristics of the basis functions. along coordinate axes coordinate axes One-dimensional basis vectors of the coordinate axes; Indicates the outer product; Store the basis function features along coordinate axes coordinate axes One-dimensional basis vectors of the coordinate axes.

5. The method according to claim 2, characterized in that, The aircraft wing shape sample set is the input physical field; when the first neural network is trained, the latent variables specific to each aircraft wing shape sample in the aircraft wing shape sample set are obtained; The flow field sample set corresponding to the aircraft wing shape sample set is the output physical field. One aircraft wing shape sample corresponds to one flow field sample to characterize the air velocity at each spatial location point of the aircraft wing shape sample. The method further includes: The flow field sample set is input into the second neural network to be trained; When the second neural network is trained, the latent variables specific to each flow field sample in the flow field sample set are obtained; Based on the latent variables specific to each aircraft wing shape sample in the aircraft wing shape sample set and the latent variables specific to each flow field sample in the flow field sample set, a processing network is trained. The trained processing network is used to establish a nonlinear mapping relationship between the latent variables of the input physical field and the latent variables of the output physical field.

6. The method according to claim 5, characterized in that, The method further includes: Input the wing shape of the target aircraft into the trained first neural network; The trained first neural network generates latent variables specific to the wing shape of the target aircraft. Based on the latent variables specific to the wing shape of the target aircraft, M fusion coefficients specific to the wing shape of the target aircraft are determined through the trained first linear supernetwork. The aircraft wing shape sample is processed at M resolutions using the M grid basis functions of the first trained neural network to obtain the M grid basis function features of the target aircraft wing shape. Based on the M fusion coefficients specific to the target aircraft wing shape, the M grid basis function features of the target aircraft wing shape are fused to obtain the modulation features specific to the target aircraft wing shape. The common features of the target aircraft wing shape and each aircraft wing shape sample in the aircraft wing shape sample set are captured by the first network weights of the first neural network after training. Based on the modulation features specific to the target aircraft wing shape sample and the common features of the target, the predicted features of the target aircraft wing shape sample are obtained; The wing shape of the target aircraft is reconstructed using the predicted features of the target aircraft's wing shape; While keeping the values ​​of the first network weights of the first neural network frozen after training, the latent variables specific to the target aircraft wing shape are updated with the goal of minimizing the difference between the reconstruction result of the target aircraft wing shape sample and the target aircraft wing shape. After updating the latent variables specific to the target aircraft wing shape, the trained processing network is used to convert the updated latent variables specific to the target aircraft wing shape into latent variables specific to the target flow field. Based on the latent variables specific to the target flow field, the target flow field is obtained through a trained second neural network, which represents the air velocity at each spatial location point of the target aircraft wing shape.

7. A training device for a physical field neural network with a cross-scale separable grid representation, characterized in that, include: The sample input module is used to input the aircraft wing shape sample set into the first neural network to be trained. The resolution allocation module is used to assign M resolutions to the M grid basis functions of the first neural network, with one grid basis function corresponding to one resolution; the M resolutions are used to balance global feature learning and local feature learning on the aircraft wing shape sample set. The feature extraction module is used to capture the common features among the various aircraft wing shape samples in the aircraft wing shape sample set by using the first network weights of the first neural network. The coefficient determination module is used to determine M fusion coefficients specific to each aircraft wing shape sample through the first neural network. The sample processing module is used to process the aircraft wing shape sample at M resolutions using the M grid basis functions to obtain the M grid basis function features of the aircraft wing shape sample; The feature fusion module is used to fuse the M grid basis function features of the aircraft wing shape sample according to the M fusion coefficients specific to the aircraft wing shape sample, so as to obtain the modulation features specific to the aircraft wing shape sample. The feature prediction module is used to obtain the predicted features of the aircraft wing shape sample based on the modulation features specific to the aircraft wing shape sample and the common features. The sample reconstruction module is used to reconstruct the wing shape sample of each aircraft using the predicted features of the sample. The parameter update module is used to update the values ​​of the first network weights of the first neural network and the parameter values ​​of the M grid basis functions based on the difference between the reconstruction result of each aircraft wing shape sample and the aircraft wing shape sample.

8. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the physical field neural network training method according to any one of claims 1-6 for cross-scale separable grid representation.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the computer program implements the steps of the physical field neural network training method for cross-scale separable grid representation as described in any one of claims 1-6.

10. A computer program product, comprising a computer program, characterized in that, When executed by a processor, the computer program implements the steps of the physical field neural network training method for cross-scale separable grid representation as described in any one of claims 1-6.

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