Complex physical simulation method based on graph neural network

Abstract

The invention provides a complex physical simulation method based on a graph neural network, and provides a dual-scale message passing method based on the graph neural network for a grid graph of a physical system. An original grid chart is mapped into a plurality of coarse-grained grid charts, and the grid charts can be processed in parallel. According to the method, each coarsened grid chart is subjected to independent down-sampling, and up-sampling is carried out uniformly; according to the method, the message transmission efficiency of the graph neural network is effectively improved, and the prediction precision is remarkably improved. Compared with an existing method based on a graph neural network, preliminary evaluation on a synthetic data set and a real data set shows that a more accurate method is provided for simulating a physical system.

Description

Technical Field

[0001] The invention relates to the field of physical simulation. Background Art

[0002] The simulation of complex physical systems is important in many engineering fields, such as structural mechanics, aerodynamics, electromagnetism, and acoustics. Traditional physical system simulation relies on the numerical solution of partial differential equations (PDE). The widely used finite element method (FEM) discretizes the physical simulation input object into a finite element mesh G, which contains a number of mesh cells, each of which consists of a number of mesh nodes and mesh edges. The finite element method is based on the mesh G, and under certain physical constraints, it performs an iterative solution to obtain the physical simulation output at each node or cell. However, the finite element method has the problems of high computational cost and long running time, and when the physical constraints change, the entire simulation process needs to be restarted. For example, Figure 1 The input mesh G is a steering wheel. The physical constraints include: the steering column is fixed on the bottom plane; an external force F is applied to the spokes. The physical simulation output is the stress value of each node on the steering wheel.

[0003] Figure 1 The figure shows a steering wheel with an external force F applied to the spokes. The stress values ​​of each node of the steering wheel after the external force F is applied are calculated using traditional PDE solvers (such as ABAQUS and LS-DYNA). However, the PDE-based simulation method is time-consuming and computationally expensive. For example, it takes more than several hours to simulate a steering wheel. In addition, when the simulation model changes slightly, the PDE solver must be re-simulated, which results in very high operating costs. Therefore, traditional methods limit their application in simulation. Summary of the invention

[0004] In order to overcome the shortcomings of the prior art, the present invention discloses a complex physical simulation method based on graph neural network, which can be used to quickly solve the physical response results of engineering parts under the influence of external factors.

[0005] Technical solution:

[0006] A complex physical simulation method based on graph neural network, characterized by adopting an encoder-processor-decoder framework;

[0007] At the encoder stage: the encoder is G 0 Encode the node attributes and edge attributes of all (L+1) graphs and convert them into embedding vectors;

[0008] At the processor stage: First, in G 0To pass messages, the model 0 Perform downsampling operations to generate L coarsened graphs G online 1 To G L ; Then, the model in the same graph G 1 To G L Messages are passed in parallel between nodes in G; Next, the model performs upsampling operations on all 91+L) graphs and updates G 0 The embedding vector of each node in G 0 Conduct messaging;

[0009] In the decoder stage: the decoder converts G 0 The node embedding vector is decoded into the final physical response result Y.

[0010] The present invention uses graph neural networks to extract geometric features and external loads of complex physical systems, and accurately defines topological relationships to achieve simulation. The present invention is oriented to the grid graph of the physical system, and proposes a dual-scale message passing method based on graph neural networks. The original grid graph is mapped into multiple coarse-grained grid graphs, which can be processed in parallel. The present invention performs separate downsampling on each coarsened grid graph and uniform upsampling; this method effectively improves the message passing efficiency of the graph neural network and significantly improves the prediction accuracy. Compared with existing graph neural network-based methods, preliminary evaluations of the present invention on synthetic data sets and real data sets show that the present invention provides a more accurate method for simulating physical systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 Traditional complex physical system simulation system solution architecture.

[0012] Figure 2 The model architecture designed by the method of the present invention.

[0013] Figure 3 Schematic diagram of downsampling and upsampling. DETAILED DESCRIPTION

[0014] To address the common problems faced by traditional numerical simulation methods, researchers have begun to explore the use of deep learning techniques to establish end-to-end approximate functions between simulation input and output. These learning-based methods, including convolutional neural networks (CNNs) and graph neural networks (GNNs), significantly reduce computational costs while maintaining high accuracy. In particular, GNNs perform well in the simulation of complex and irregular topological structures, and develop simulation methods with high accuracy and high computational efficiency. These methods greatly simplify the simulation process of complex physical systems and show higher adaptability in a variety of application scenarios. For example, after the physical system is discretized into a grid, MeshGraphNets can rely on the powerful representation ability of GNNs to learn physical properties from the input grid graph. Nevertheless, in the segmented grid structure, the degree distribution of nodes is relatively regular and significantly different from the power-law distribution commonly seen in Internet social networks. Therefore, MeshGraphNet only performs message passing on single-scale graphs, which is not effective and difficult to achieve good convergence on these grid graphs. By adopting a hierarchical message passing strategy, Multiscale MeshGraphNets utilizes a multi-level graph network structure of fine-grained and coarse-grained mesh graphs to significantly improve the accuracy of physical simulation results.

[0015] Inspired by the above work, the present invention studies the simulation application of complex physical systems in real-world product design. The present invention aims to generate the response results of a given physical object under the action of external factors. Consider the scene in the figure, where the vehicle steering wheel is acted upon by an external force F. Figure 2 : The left side image shows the segmented meshes, and the right side image shows the effective stress heat map on these meshes. If the effective stress exceeds a certain threshold, the steering wheel may be distorted or even broken. Through the heat map, engineers can effectively identify the weak points of the steering wheel, and then optimize the design and enhance the structural strength.

[0016] The above-mentioned previous works perform poorly in the above-mentioned scenarios for the following reasons. First, these research results are mainly developed for simulation methods based on time steps or time series, in which a single grid can be intuitively regarded as a time series of simulation results over multiple time steps. Therefore, the simulation can essentially be regarded as a multivariate time series prediction problem, by using historical simulation results as training data to predict the simulation results at future time steps. The present invention is not concerned with intermediate results, but with the final simulation output results under the action of physical objects and external factors (i.e., inputs). Secondly, in terms of grid partitioning, previous works require pre-division of multiple layers of fine-grained and coarse-grained grids to construct a hierarchical graph network. In fact, high-quality partitioning of both fine-grained and coarse-grained grids requires considerable overhead. Finally, in time step simulation, the grid graph remains fixed during the simulation process, and the prediction is heavily dependent on the temporal pattern of the underlying physical dynamics.

[0017] On the contrary, in the case of the present invention, different physical objects have different grid diagrams, and the key to the present invention is how to adapt the present invention to various grid diagrams and external factors. Therefore, the present invention can accurately predict the system response under various physical objects (corresponding to grid diagrams) and external factor input conditions.

[0018] The technical solution provided by the present application will be further described below in conjunction with specific embodiments and accompanying drawings. The advantages and features of the present application will become more apparent with the following description.

[0019] Problem definition:

[0020] The present invention adopts the widely used finite element method (FEM) to numerically solve partial differential equations (PDEs) and simulate the studied scenarios. The numerical results obtained by the PDE solver can be used as the reference truth value. The present invention discretizes the input physical object P into a set of two-dimensional or three-dimensional grids to maintain consistency with the finite element grid generation. In essence, these grids are adjacent finite elements used to approximate the geometric shape of the physical object. Given a discretized grid, the present invention represents the physical object by inputting a grid graph G = {V, E}, which includes a node set V = {v i} and edge set E = {e i,j}, node v i and v j Each node is connected to each other by edges. In addition, each node has a three-dimensional (or two-dimensional) coordinate (or ).

[0021] The present invention is achieved by F = {f k} to represent external factors, where node k is a node in the node set V. Examples of external factors in practical applications can be gravity, wind pressure, vibration, or other physical forces. Given the grid graph G and external force F, the model output generates a d-dimensional physical response result Y = {y i}, where node i is a node in the node set V. The d-dimensional response result Y is usually represented by a numerical vector, such as effective stress and displacement. Therefore, the model focuses on learning a regression model.

[0022] Thus, the input of the present invention is the physical system O simulated by the grid graph G = (V, E) and the external force F = {f k}, the output is the physical response result Y={y i}.

[0023] A complex physics simulation method based on graph neural network adopts the basic encoder-processor-decoder framework (see Figure 1 ).

[0024] At the encoder stage: the encoder is G 0 The node attributes of the graph and the edge attributes of all (L+1) graphs are encoded and converted into embedding vectors.

[0025] At the processor stage: First, in G 0 To pass messages, the model 0 Perform downsampling operations to generate L coarsened graphs G online 1 To G L Then, the model in the same graph G 1 To G L Next, the model performs upsampling operations on all (1+L) graphs and updates G 0 The embedding vector of each node in G 0 Messaging is performed.

[0026] In the decoder stage: the decoder converts G 0 The node embedding vector is decoded into the final physical response result Y.

[0027] The following is a detailed description of each design part in the process framework.

[0028] 1. Encoder design method

[0029] The model encodes the graph G = {V, E} and the external force F to generate the initial graph G 0 The node embedding vector V 0 and the edge embedding vector e that needs to be processed in the subsequent model i,j , e i,j It may be the same figure G 0 ,G 1 ,...,G L The edges in the input graph G 0 With the coarsened graph G 1 ,...,G L The crossing edges between V 0 Each node i in is encoded by the node encoding function ε v Encode the external force f corresponding to the node i , generate the corresponding node embedding vector Similarly, each edge Through the edge encoding function ε e Encode the coordinates x of the two nodes i and j corresponding to the edge i and x j The displacement between i,j =x i -x j and its distance |d i,j |(The present invention uses Euclidean distance) to generate the corresponding edge embedding vector ei,j . The embedding function ε of nodes and edges v and ε e This is achieved through multi-layer perceptrons (MLPs) with the following formula:

[0030]

[0031] 2. Processor Design Method

[0032] The processor consists of three parts. First, it is a graph G at different scales. l ,l=0,1,…,L, then the message is passed. Then, it includes two processes: downsampling and upsampling. Among them, downsampling converts the input grid graph G 0 Coarsened into a parallel coarsened graph G 1 ,G 2 ,…,G L ; Upsampling through the set G 1 ,G 2 ,…,G L Restore Update G 0 .

[0033] Message passing: In Figure G l In the example, message passing first passes through the edge embedding vector and the node embedding vectors of nodes i and j at both ends of the edge and To update the embedding vector of the edge in the graph Then, by aggregating the embedding vectors of the edges and the embedding vectors of the nodes themselves To update the embedding vector of the node. The update function of the edge and node is passed through f E and f V It means that the present invention implements the update function through multi-layer perceptrons (MLPs), and the update equations of edges and nodes are as follows:

[0034]

[0035] Downsampling and upsampling ( Figure 3 In order to explain the details of the method more clearly, we simplify the model by assuming that L = 2. The model has an input grid map G 0 And two coarsened graphs G 1 and G 2 , the coarsened graph approximates the input grid graph using a smaller number of nodes and edges. For example, in Figure 3 In (a), the model converts G 0 The grid nodes in the grid are divided into four clusters A1, B1, C1, and D1, and then the cluster center nodes a1, b1, c1, and d1 corresponding to the four clusters are used to represent these clusters. 0There is at least one edge connecting the nodes of the two clusters, then the nodes corresponding to the two clusters are connected. Similarly, Figure 3 In (b), the model has a coarsened graph G with 5 nodes. 2 Now, if Figure 3 As shown in (c), a node (such as n∈G 0 ) belong to two different clusters B1 and C2.

[0036] Given an input graph G 0 And the above L = 2 coarsened graphs G 1 and G 2 , the model performs downsampling and upsampling operations. For a certain cluster, the model downsamples from all cluster member nodes to the cluster center node of the cluster. Conversely, the model upsamples from the cluster center node to the cluster member nodes. For example, in Figure 3 In (a), the 7 nodes in cluster B1 (including node n) are downsampled to the cluster center node b1, and then reversely upsampled from b1 to these 7 nodes. Figure 3 In (c), node n receives two upsampling messages from b1 and c2.

[0037] For each coarsened graph G l , the present invention considers G 0 and G l The edge E between 0,l .side Connect source nodes k ∈V 0 and the target node r k ∈V l In order to aggregate from G 0 To G l Node embedding, that is, downsampling, the present invention uses node V 0 and edge E 0,l Update V l Node embedding of V,down and f E,down are the aggregation functions of node and edge embeddings, respectively. These two functions are implemented by multi-layer perceptrons (MLPs) as follows:

[0038]

[0039] Finally, the model embeds nodes from multiple subgraphs G by upsampling l Spread to G 0 , that is, upsampling. Define two functions f V,up and f E,up To update the embedding vectors of nodes and edges, these two functions are implemented by multi-layer perceptrons (MLPs). The formula is as follows:

[0040]

[0041] 3. Decoder Design Method

[0042] Decoder Y = (V 0 ) embeds the node into a vector Through the function δ v Decoded into output feature y i , using multi-layer perceptrons (MLPs) to implement the function δ v , the formula is as follows:

[0043]

[0044] IV. Experimental Verification

[0045] When training the graph network, the present invention first normalizes all simulated input and output data to a mean of 0 and a variance of 1. The present invention uses an L2 loss function to measure the error between the predicted value and the true value, and uses an Adam optimizer to update the network parameters.

[0046] Dataset: The present invention uses a generated dataset and a dataset of a real application scenario in the table for performance evaluation. All datasets contain the input geometry P and external force F, generating the stress field physical response result Y at the node. For each dataset, 60% of the samples are used for training, 20% for validation, and 20% for testing.

[0047] Cantilever beam: The popular PDE solver ABAQUS is used to generate a cantilever beam dataset. The cantilever beam is a two-dimensional rectangular structure with a length of x = 100 mm and a width of y = 15 mm. In addition, there is a circular hole with a radius of R = 2.5 mm in the cantilever beam. By changing the center position of the hole (c x ,c y ), the present invention obtains 111=3*37 beam structures, that is, (c x ,c y )=(5+2.5*i,5+2.5*j), where i=0,1,2 and j=0,1,…,36. The present invention applies a force F of 255 Newtons at the end of the beam in the opposite direction of the y-axis. For a single cantilever beam, the present invention discretizes its surface into a 2D triangular mesh using the mesh generation tool provided by ABAQUS, and performs PDE simulation to generate the true value of stress.

[0048] Real Steering Wheel: The present invention uses a real data set containing 135 significantly different steering wheel structures provided by an automotive parts supplier. The present invention applies an external force F of 700 Newtons in the center of the steering wheel spoke in the opposite direction of the z-axis (see table). LS-DYNA is used to simulate the torsion test and measure the stress field in the steering wheel. Here, the generated mesh includes three mixed types (hexahedron, pentahedron and tetrahedron).

[0049] The present invention is compared with three recent state-of-the-art techniques.

[0050] (1) MeshGraphNets (MGN): This work only requires one level of fine mesh graph, and all message passing is performed on the fine mesh graph.

[0051] (2) Multiscale MeshGraphNets (MS-MGN): This work requires the generation of a fine mesh graph and a coarse mesh graph through a traditional solver in advance, forming a two-level graph network. Message passing involves down / up sampling across the two-level graph network.

[0052] (3) MultiscaleGNN implements multi-level hierarchical graph networks, involving down / up sampling message passing across these graph networks. MS-MGN, MultiscaleGNN and the present invention all involve multi-level GNNs. Therefore, for fairness, the present invention and MS-MGN, MultiscaleGNN set up two levels of GNNs, while MGN maintains the original method and sets up one level of GNN.

[0053] The present invention uses the predicted results and CAE simulation results to settle errors and measure the effects of all methods. The present invention uses the mean square error (MSE) to measure the effects of different models. The comparison results are as follows:

[0054] Datasets MGN MS-MGN MultiscaleGNN Ours Beam 257.5341 13.9726 9.3295 7.2401 Steering Wheel 246.8273 - 117.1109 46.1042

[0055] In the table, the present invention performs a benchmark study of four methods on two datasets. First, the prediction error of the present invention on all datasets is significantly lower than that of the other three control groups. Unlike MS-MGN, the present invention only needs one input grid map for a specific physical object, which shows that the present invention has advantages over MS-MGN and MultiscaleGNN in practical applications.

[0056] The above description is only a description of the preferred embodiments of the present application, and is not intended to limit the scope of the present application. Any changes or modifications made by any person skilled in the art based on the above disclosed technical contents shall be deemed as equivalent effective embodiments and shall fall within the scope of protection of the technical solution of the present application.

Claims

1. A complex physical simulation method based on graph neural network, characterized in that: Adopt encoder-processor-decoder framework; At the encoder stage: the encoder is G 0 Encode the node attributes and edge attributes of all (L+1) graphs and convert them into embedding vectors; At the processor stage: First, in G 0 To pass messages, the model 0 Perform downsampling operations to generate L coarsened graphs G online 1 To G L ; Then, the model in the same graph G 1 To G L Messages are passed in parallel between nodes in the Next, the model upsamples all (1+L) graphs and updates G 0 The embedding vector of each node in G 0 Conduct messaging; In the decoder stage: the decoder converts G 0 The node embedding vector is decoded into the final physical response result Y.

2. The complex physical simulation method based on graph neural network according to claim 1, characterized in that: The encoder design method is: The model encodes the graph G = {V, E} and the external force F to generate the initial graph G 0 The node embedding vector V 0 and the edge embedding vector e that needs to be processed in the subsequent model i,j ;e i,j It is the same graph G 0 ,G 1 ,...,G L The edges in the graph G 0 With the coarsened graph G 1 ,...,G L The intersection between V 0 Each node i in is encoded by the node encoding function ε v Encode the external force f corresponding to the node i , generate the corresponding node embedding vector Similarly, each edge Through the edge encoding function ε e Encode the coordinates x of the two nodes i and j corresponding to the edge i and x j The displacement between i,j =x i -x j and its distance |d i,j |, generate the corresponding edge embedding vector e i,j ; Embedding functions ε of nodes and edges v and ε e This is achieved through multi-layer perceptrons (MLPs) with the following formula:

3. The complex physical simulation method based on graph neural network according to claim 1, characterized in that: The design method of the processor is: The processor consists of three parts. First, it is a graph G at different scales. l ,l=0,1,…,L, then, it includes two processes: downsampling and upsampling. Among them, downsampling is the input grid graph G 0 Coarsened into a parallel coarsened graph G 1 ,G 2 ,…,G L ; Upsampling through the set G 1 ,G 2 ,…,G L Restore Update G 0 .

4. The complex physical simulation method based on graph neural network as claimed in claim 3, characterized in that: Messaging: In Figure G l In the example, message passing first passes through the edge embedding vector and the node embedding vectors of nodes i and j at both ends of the edge and To update the embedding vector of the edge in the graph Then, by aggregating the embedding vectors of the edges and the embedding vectors of the nodes themselves To update the embedding vector of the node; the update function of the edge and node is through f E and f V Indicates that the update function is implemented by multi-layer perceptrons (MLPs), and the update equations for edges and nodes are as follows:

5. The complex physical simulation method based on graph neural network as claimed in claim 3, characterized in that: Downsampling and upsampling: For each coarsened graph G l , consider G 0 and G l The edge E between 0,l ,side Connect source nodes k ∈V 0 and the target node r k ∈V l ; In order to aggregate from G 0 To G l Node embedding, that is, downsampling, using node V 0 and edge E 0,l Update V l Node embedding of V,down and f E,down are the aggregation functions of node and edge embeddings, respectively. These two functions are implemented by multi-layer perceptrons (MLPs) as follows: Finally, the model embeds nodes from multiple subgraphs G by upsampling l Spread to G 0 , that is, upsampling; define two functions f V ,up and f E,up To update the embedding vectors of nodes and edges, these two functions are implemented by multi-layer perceptrons (MLPs). The formula is as follows:

6. The complex physical simulation method based on graph neural network according to claim 1, characterized in that: The design method of the decoder is: Decoder Y = (V 0 ) embeds the node into a vector Through the function δ v Decoded into output feature y i , using multi-layer perceptrons (MLPs) to implement the function δ v , the formula is as follows:

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