A physical system trajectory prediction method and system based on an equivariant graph neural network

Abstract

The application discloses a physical system trajectory prediction method and system based on an isometric graph neural network, and relates to the technical field of trajectory prediction. The method comprises the following steps: constructing a virtual node corresponding to a real fluid particle node according to a target state at a given moment, and constructing a trajectory prediction model based on an isometric graph neural network according to the virtual node; the target state comprises the features and coordinates of a target fluid particle at the given moment; based on the trajectory prediction model, the state data of the real fluid particle node is first updated, and then the state data of the virtual node is updated according to the updated state data. The application can improve the prediction accuracy of the trajectory of the physical system.

Description

Technical Field

[0001] This invention relates to the field of trajectory prediction technology, and in particular to a method and system for predicting the trajectory of a physical system based on an equivariant graph neural network. Background Technology

[0002] In the field of complex physical system simulation, trajectory prediction in fluid dynamics, molecular dynamics, and multibody interaction systems remains a core problem. Traditional methods typically rely on numerical methods such as finite element method, finite difference method, or smoothed particle hydrodynamics (SPH) to characterize physical laws. While these methods can guarantee a certain level of accuracy, their computational complexity is extremely high, making it difficult to balance efficiency and scalability for large-scale, long-term simulations. In recent years, with the development of deep learning technology, researchers have attempted to accelerate physical system simulations using neural networks. For example, methods based on convolutional neural networks (CNNs) can predict flow field evolution on fixed grids, but due to their reliance on regular grid structures, they struggle to handle complex boundaries and irregular geometries. On the other hand, methods based on graph neural networks (GNNs) learn dynamic laws by constructing adjacency relationships between particles. Representative works such as Interaction Networks and Graph Network Simulator (GNS) have shown good prediction performance in fluid and multibody dynamics tasks. However, these methods still have significant drawbacks: CNN methods are limited by the grid and lack flexibility; traditional GNN methods do not explicitly introduce physical symmetry modeling, so their generalization ability is insufficient in symmetrical scenarios such as rotation, translation, and reflection; in the prediction of complex three-dimensional flow fields, ordinary neural network methods often cannot guarantee physical consistency and are prone to energy non-conservation or numerical drift.

[0003] To overcome the aforementioned problems, researchers proposed the Equivariant Graph Neural Network (EGNN). By introducing geometric constraints and tensor operators into the message-passing mechanism, it preserves symmetries such as translation and rotation, achieving significant progress in tasks such as molecular modeling and dynamic prediction. However, EGNNs typically require high-order tensor product operations in multibody interaction modeling, which is extremely computationally expensive, making them difficult to apply in large-scale fluid particle simulations. Therefore, existing methods are either computationally inefficient or lack physical consistency and generalization ability, failing to balance efficiency and accuracy. Summary of the Invention

[0004] The purpose of this invention is to provide a method and system for predicting the trajectory of a physical system based on an equivariant graph neural network, aiming to solve or improve at least one of the above-mentioned technical problems.

[0005] To achieve the above objectives, the present invention provides the following solution: A method for predicting the trajectory of a physical system based on an isotropic graph neural network includes: Based on the target state at a given time, virtual nodes corresponding to real fluid particle nodes are constructed, and a trajectory prediction model based on an isotropic graph neural network is constructed based on the virtual nodes; the target state includes the characteristics and coordinates of the target fluid particle at a given time. Based on the trajectory prediction model, the state data of the real fluid particle nodes are first updated, and then the state data of the virtual nodes are updated according to the updated state data.

[0006] Optionally, the process for calculating the state data of the virtual node is as follows: The edge features and distance between any two real fluid particle nodes are calculated based on the four-point positioning principle, and then processed by a multilayer perceptron. The edge features are encoded and then processed by a multilayer perceptron. and The initialization features of the virtual nodes are calculated, and finally, the aggregate information of all neighborhoods for each real fluid particle node is calculated to obtain the coordinates of the virtual nodes.

[0007] Optionally, the formula for calculating the state data of the virtual node includes: in, This represents the encoded edge features. , and This represents the multilayer perceptron operation. Table Node i Features shown, Represents a node j Features Indicates the characteristics of the connected edges. Indicates the distance between two nodes. This indicates the shared characteristics of virtual nodes. Represents the coordinates of the virtual node. This represents a vector consisting entirely of 1s. Represents the coordinates of the actual node. Represents a node i The neighborhood, α i φ represents the normalization coefficient. Z This represents a multilayer perceptron used to calculate coordinate difference weights.

[0008] Optionally, the process of updating the state data of the real fluid particle nodes is specifically represented as follows: in, This represents the encoded edge message. and This represents the multilayer perceptron operation. Table Node i Features shown, Represents a node j Features Indicates the characteristics of the connected edges. Indicates the distance between two nodes. α i Represents the normalization coefficient. Represents a node i The neighborhood, Vector messages representing connected edges This represents a multilayer perceptron for calculating vector feature weights. This represents the aggregated scalar message. This represents the aggregated vector message. Represents the coordinates of the node.

[0009] Optionally, updating the virtual node's state data based on the updated state data specifically involves: Based on the updated state data, an equivalent DeepSet is constructed, the coordinate difference between the virtual node and the real fluid particle node is calculated, and a novel multilayer perceptron is introduced. Constructing complete scalar information Finally, the node features and coordinates are updated sequentially.

[0010] Optionally, the process of updating the state data of the virtual node is specifically represented as follows: in, This represents the coordinate difference between the virtual node and the real node. This represents a vector consisting entirely of 1s. Represents the coordinates of the virtual node. Represents scalar information. This indicates the shared characteristics of virtual nodes. Represents a node j Features Table Node i Features shown, α i Represents the normalization coefficient. Represents a node i The neighborhood, Represents the coordinates of neighboring nodes. This represents a complete multilayer perceptron used for edge message computation. This represents a complete multilayer perceptron used for updating the scalar features of real nodes. This represents a complete multilayer perceptron used for updating the scalar features of virtual nodes. Represents the coordinates of the actual node. (v) This represents a multilayer perceptron that utilizes weights derived from information from virtual nodes to update the coordinates of real nodes. (v) This represents a multilayer perceptron that utilizes information from virtual nodes and the weights used to update the coordinates of virtual nodes.

[0011] The present invention also provides a physical system trajectory prediction system based on an isovariant graph neural network, comprising: The model building unit is used to construct virtual nodes corresponding to real fluid particle nodes based on the target state at a given time, and to construct a trajectory prediction model based on an isotropic graph neural network based on the virtual nodes; the target state includes the features and coordinates of the target fluid particle at a given time. The trajectory prediction unit is used to first update the state data of the real fluid particle nodes based on the trajectory prediction model, and then update the state data of the virtual nodes according to the updated state data.

[0012] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects: This invention discloses a method and system for predicting the trajectory of a physical system based on an isotropic graph neural network. The method includes constructing virtual nodes corresponding to real fluid particle nodes based on the target state at a given time, and constructing a trajectory prediction model based on the isotropic graph neural network based on the virtual nodes. The target state includes the characteristics and coordinates of the target fluid particle at a given time. Based on the trajectory prediction model, the state data of the real fluid particle nodes are first updated, and then the state data of the virtual nodes are updated according to the updated state data. This invention can improve the accuracy of predicting the trajectory of a physical system. This invention first introduces a geometric graph data structure to model the physical system. In the modeling process, a set of reference points is learned for each particle node. The coordinates of other particles are transformed into distance vector representations through the four-point positioning principle, and a deep ensemble network is used to efficiently process the set of distance vectors corresponding to all particle coordinates, thereby effectively modeling complex multibody interactions while maintaining physical symmetry. This method can balance physical consistency, prediction accuracy, and computational efficiency, breaking through the bottleneck of existing technologies and has strong practical application value. Attached Figure Description

[0013] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0014] Figure 1This is a schematic diagram of the physical system trajectory prediction method based on equivariant graph neural network of the present invention; Figure 2 This is a schematic diagram showing the position and state of each physical entity in three-dimensional space in this embodiment. Detailed Implementation

[0015] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0016] The purpose of this invention is to provide a method and system for predicting the trajectory of a physical system based on an equivariant graph neural network, aiming to solve or improve at least one of the above-mentioned technical problems.

[0017] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0018] like Figure 1 As shown, this invention provides a method for predicting the trajectory of a physical system based on an isovariant graph neural network. First, virtual nodes are constructed around each node using learnable weights. Second, using the four-point positioning principle, the spatial coordinates of physical entities are transformed into distance vectors from their virtual nodes, and encoded using a DeepSet to obtain the standard form of the geometric graph. This standard form is then embedded into the scalar channel of the neural network, thereby constructing the isovariant graph neural network. This method can effectively model multi-body interactions in complex systems while maintaining physical symmetry, achieving efficient trajectory prediction for physical systems. Specific steps include: Step 100: Construct virtual nodes corresponding to real fluid particle nodes based on the target state at a given time, and construct a trajectory prediction model based on an isotropic graph neural network based on the virtual nodes; the target state includes the characteristics and coordinates of the target fluid particle at a given time.

[0019] Step 200: Based on the trajectory prediction model, first update the state data of the real fluid particle nodes, and then update the state data of the virtual nodes according to the updated state data.

[0020] As a specific implementation method, the following specific processing procedure is provided.

[0021] First, regarding the graph neural network used to construct the trajectory prediction model in this embodiment, a graph is an important data structure for describing objects and their relationships, consisting of nodes and edges. Nodes represent basic entities in a physical system, such as particles, molecules, or rigid bodies; edges represent relationships between entities and can have weights to represent the strength, distance, or dependency of interactions. In physical systems, interactions between nodes are usually bidirectional; therefore, the system is often represented as an undirected graph to describe the symmetry relationships and interaction laws between particles. This graph structure transforms the trajectory prediction problem of a physical system into a learning and inference task based on graph neural networks, thus providing an efficient and physically consistent method for modeling and simulating large-scale complex physical systems. The entire physical system can be represented as a geometric graph, that is, introducing geometric information (such as coordinates, velocity, and other physical properties) on top of the traditional graph structure. Specifically, it can be represented as... ,in Representing the system A physical entity (such as a particle, molecule, or rigid body). It is an adjacency matrix (0-1 matrix) used to represent the interaction relationships between physical entities, such as This explains the first in the system The physical entity and the first These physical entities can interact and exchange information. represent The coordinate information of a physical entity in three-dimensional space. Optionally, when the dataset contains velocity information, velocity attributes can also be added to the node features of the geometry, similar to coordinate attributes, to more completely describe the dynamic state of the physical system.

[0022] The trajectory prediction task of a physical system requires predicting the motion trajectory of each physical entity in the system in three-dimensional space. This process needs to satisfy the equivariance property of E(3). Here, E(3) represents a special Euclidean group in three-dimensional space, used to describe rigid body transformations in three-dimensional space, including translation and rotation. The so-called equivariance means that the prediction result of the physical system will change accordingly with the input rigid body transformation, thus ensuring that the prediction result is independent of the choice of reference coordinate system. In contrast, invariance means that the prediction result remains completely consistent under different coordinate systems, while equivariance requires that the output changes accordingly with the transformation of the input. This equivariance is of great significance in the trajectory prediction of physical systems, because the physical laws themselves should remain consistent under spatial transformations, thus ensuring the physical consistency and generalization ability of the prediction. Existing E(3) equivariant graph neural networks (such as the EGNN model) can make full use of this property when dealing with tasks such as molecular dynamics, multibody systems and fluid dynamics. The model proposed in this invention also belongs to the E(3) equivariant graph neural network. By encoding the geometric information of the nodes, it can achieve high-precision prediction of the trajectory of the physical system, while ensuring the correctness of the prediction results under translation and rotation transformations.

[0023] like Figure 2 As shown, the task of physical system trajectory prediction is to predict the position and state of each physical entity in the system in three-dimensional space. Assuming that the motion direction of a certain physical entity in the system is the initial orientation, when the spatial coordinates of the entity are rotated as a whole (for example, rotated by 90°), the motion direction predicted by the model will also rotate accordingly, thus ensuring that the prediction result is consistent with the rotation of the input coordinate system. This reflects the E(3) equivariance of the model.

[0024] The core of physical system trajectory prediction lies in predicting the state of each physical entity at the next moment based on the system state at a given moment. This process can generally be divided into the following three steps: 1) Construct virtual nodes based on each fluid particle node, using the following formula: Represents the coordinates of the virtual node. Represent the shared features of virtual nodes and calculate the coordinate difference between the node and the virtual node: Specifically, the formula in the first row uses the features of node i. Features of node j and edge features and distance After passing through a multilayer sensor Encoding yields edge features In the second line of the formula, a multilayer perceptron is further introduced. and Calculate the initialization characteristics of the virtual nodes; in the third line of the formula, Represent a vector of all 1s, and then for the neighborhood of node i... All the information is aggregated to obtain the coordinates of the virtual node. .

[0025] 2) Similar to a normal EGNN, update the features and coordinates of all real fluid particle nodes: All symbols that are the same as those in the formula of step 1 have the same meaning: normalization coefficient. Representing the neighborhood The number of nodes in the system.

[0026] 3) Further update the features and coordinates of the fluid particle nodes and virtual nodes. This step is equivalent to constructing a DeepSet and then obtaining the desired objective function through isovariant feature basis functions: Specifically, among them This represents the coordinate difference between virtual nodes and real nodes, and then a novel multilayer perceptron is introduced. Constructing complete scalar information Finally, the node features and coordinates are updated sequentially.

[0027] Based on the above technical solution, the following embodiment for vehicle trajectory prediction in a transportation system is provided.

[0028] For all vehicles in a region, each vehicle can be considered a node, with its vehicle type serving as a feature, along with corresponding coordinates and speed. Since a vehicle's route is generally only influenced by nearby vehicles, a distance-based edge construction method can be used to build the geometric graph. By inputting the aforementioned information, the positions of vehicles at subsequent time points can be predicted, thus achieving the task of predicting vehicle trajectories in the traffic system.

[0029] In addition, this embodiment also uses two datasets, 5-body and Water-3D mini, to test the effectiveness of this technical solution. On each dataset, the performance of the model and baselines such as EGNN, FastEGNN, and HEGNN was tested under different settings of 1-4 layers.

[0030] 5-body is a commonly used dataset for evaluating trajectory prediction in physical systems. Specifically, it is a dataset generated by simulation. In the simulation of this example, each system contains 5 charged particles, each with a random charge value of ±1, and their motion is driven by Coulomb forces. The dataset is divided into: a training set of 5000 samples, a validation set of 2000 samples, and a test set of 2000 samples. The task objective is to predict the positions of all particles after 1000 time steps. The results are shown in Table 1.

[0031] Table 1

[0032] Water-3D is a large-scale particle-based fluid simulation dataset generated using Smoothed Particle Hydrodynamics (SPH). It records the dynamic process of water falling into boxes. Each system contains an average of 7806 particles and 94999 edges, with an edge truncation value of 0.035. Due to the large sample size of the original dataset, this embodiment limits the evaluation scope to a 1 / 5 subset, constructing a new evaluation metric, Water-3D mini, which contains 3000 / 300 / 300 samples from the original dataset (15,000 / 1,500 / 1,500) for training, validation, and testing. The maximum training epoch is set to 1,000. The results are shown in Table 2.

[0033] Table 2

[0034] As can be seen from the above, the model in this embodiment outperforms the traditional baseline EGNN, FastEGNN, and HEGNN in both tasks. Furthermore, the model in this embodiment, with a 1-2 layer setup, performs close to the performance of other models with 3-4 layers, fully demonstrating the effectiveness of this modeling method for fluid simulation tasks.

[0035] 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.

[0036] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for predicting the trajectory of a physical system based on an isotropic graph neural network, characterized in that, include: Based on the target state at a given time, virtual nodes corresponding to real fluid particle nodes are constructed, and a trajectory prediction model based on an isotropic graph neural network is constructed based on the virtual nodes; the target state includes the characteristics and coordinates of the target fluid particle at a given time. Based on the trajectory prediction model, the state data of the real fluid particle nodes are first updated, and then the state data of the virtual nodes are updated according to the updated state data. The process for calculating the state data of the virtual node is as follows: The edge features and distance between any two real fluid particle nodes are calculated based on the four-point positioning principle, and then processed by a multilayer perceptron. The edge features are encoded and then processed by a multilayer perceptron. and The initialization features of the virtual nodes are calculated, and finally, the aggregate information of all neighborhoods of each real fluid particle node is calculated to obtain the coordinates of the virtual nodes. The formula for calculating the state data of the virtual node includes: in, This represents the encoded edge features. , and This represents the multilayer perceptron operation. Represents a node i Features Represents a node j Features Indicates the characteristics of the connected edges. Indicates the distance between two nodes. This indicates the shared characteristics of virtual nodes. Represents the coordinates of the virtual node. This represents a vector consisting entirely of 1s. Represents the coordinates of the actual node. Represents a node i The neighborhood, Represents the normalization coefficient. This represents a multilayer perceptron used to calculate coordinate difference weights.

2. The physical system trajectory prediction method based on equivariant graph neural networks according to claim 1, characterized in that, The process of updating the state data of the real fluid particle nodes is specifically represented as follows: in, This represents the encoded edge features. and This represents the multilayer perceptron operation. Represents a node i Features Represents a node j Features Indicates the characteristics of the connected edges. Indicates the distance between two nodes. Represents the normalization coefficient. Represents a node i The neighborhood, Vector messages representing connected edges This represents a multilayer perceptron for calculating vector feature weights. This represents the aggregated scalar message. This represents the aggregated vector message. Represents the coordinates of the actual node.

3. The physical system trajectory prediction method based on equivariant graph neural networks according to claim 1, characterized in that, The step of updating the virtual node's state data based on the updated state data specifically involves: Based on the updated state data, an equivalent DeepSet is constructed. The coordinate difference between virtual nodes and real fluid particle nodes is calculated, and a novel multilayer perceptron is introduced. Constructing complete scalar information Finally, the node features and coordinates are updated sequentially.

4. The physical system trajectory prediction method based on equivariant graph neural networks according to claim 3, characterized in that, The process of updating the state data of the virtual node is specifically represented as follows: in, This represents the coordinate difference between the virtual node and the real node. This represents a vector consisting entirely of 1s. Represents the coordinates of the virtual node. Represents scalar information. This indicates the shared characteristics of virtual nodes. Represents a node j Features Represents a node i Features Represents the normalization coefficient. Represents a node i The neighborhood, Represents the coordinates of neighboring nodes. This represents a complete multilayer perceptron used for edge message computation. This represents a complete multilayer perceptron used for updating the scalar features of real nodes. This represents a complete multilayer perceptron used for updating the scalar features of virtual nodes. Represents the coordinates of the actual node. This represents a multilayer perceptron that utilizes weights derived from information from virtual nodes to update the coordinates of real nodes. This represents a multilayer perceptron that utilizes information from virtual nodes and the weights used to update the coordinates of virtual nodes.

5. A trajectory prediction system for a physical system based on an isovariant graph neural network, using the method described in any one of claims 1-4, characterized in that, include: The model building unit is used to construct virtual nodes corresponding to real fluid particle nodes based on the target state at a given time, and to construct a trajectory prediction model based on an isotropic graph neural network based on the virtual nodes; the target state includes the features and coordinates of the target fluid particle at a given time. The trajectory prediction unit is used to first update the state data of the real fluid particle nodes based on the trajectory prediction model, and then update the state data of the virtual nodes according to the updated state data.

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