Chaotic system modeling and predicting method based on graph neural network symbol regression algorithm
By constructing a symbolic expression tree using a graph neural network and optimizing the generation of differential equations, the problem of modeling and predicting chaotic systems under noise influence in existing technologies is solved, and high-precision state prediction is achieved in high-noise environments.
Patent Information
- Application Number
- CN202511653689.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-12
- Publication Date
- 2026-02-17
AI Technical Summary
Existing technologies struggle to automatically recover differential equations from noisy chaotic trajectory data and apply them to physical information neural networks for effective state prediction, especially when the governing equations are unknown.
A symbolic expression tree is constructed using a graph neural network. Feature propagation is performed through a multi-layer graph neural network. Combined with exploration reward and similarity-weighted policy gradient optimization, a valid mathematical expression is generated. The reconstructed differential equation is then embedded into the loss function of the physical information neural network for prediction.
It achieves accurate recovery of chaotic system equations under high noise conditions, improves prediction accuracy and robustness, and completes end-to-end modeling from data to prediction.
Smart Images
Figure CN121543632A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of software technology, and more specifically to a method for modeling and predicting chaotic systems based on graph neural network symbolic regression algorithm. Background Technology
[0002] Chaotic systems are a class of nonlinear dynamical systems that are extremely sensitive to initial conditions and possess inherent randomness. Although their evolutionary behavior is governed by deterministic differential equations, they exhibit complex trajectories resembling randomness. Such systems are widely found in nature and engineering practice, such as atmospheric flow, turbulence, neuronal firing, and laser dynamics. Their mathematical models are typically described by a set of coupled ordinary differential equations (ODEs) or partial differential equations (PDEs), which are concise in form but highly nonlinear in structure.
[0003] In actual observations, system states are often affected by factors such as sensor noise and environmental disturbances, leading to varying degrees of data distortion. How to automatically reconstruct the analytical expression governing system evolution from noisy, finite-length trajectory data, and further use it for state prediction or control, is a core challenge in scientific modeling and intelligent system design.
[0004] Symbolic Regression (SR) aims to automatically discover concise and interpretable mathematical expressions from data, and is an important tool for scientific discovery. Traditional methods such as Genetic Programming (GP), while capable of searching the expression space, suffer from high computational costs, premature convergence, and difficulty in recovering complex coupled structures in high-dimensional or noisy scenarios. In recent years, reinforcement learning-based methods such as Deep Symbolic Regression (DSR) have generated expressions using RNNs, but their sequence modeling approach suffers from a structural misalignment with the hierarchical structure of expression trees, leading to inefficient information transfer.
[0005] To improve the performance of symbolic regression, an attempt was made to introduce Graph Neural Networks (GNNs) into the expression generation process. However, existing GNN methods mostly use GNNs as expression evaluators or substructure scorers, without directly modeling the expression tree generation process. More importantly, current methods lack effective integration with downstream physical modeling tasks (such as state prediction).
[0006] Physical-Informed Neural Networks (PINNs) enable data-driven modeling of physical systems by embedding the governing equations into a loss function. However, PINNs rely on the knowledge of the specific form of the governing equations. For unknown chaotic systems, if the differential equations cannot be obtained a priori, effective physical constraints cannot be constructed, limiting the application of PINNs in open scenarios.
[0007] In summary, there is an urgent need for an end-to-end method that can automatically discover differential equations from noisy chaotic trajectories and use these equations to guide robust predictions of PINNs. Summary of the Invention
[0008] In view of the above problems, this invention is proposed to provide a chaotic system modeling and prediction method based on graph neural network symbolic regression algorithm to overcome or at least partially solve the above problems. The method first uses graph neural network (GCN) to construct symbolic expression tree to accurately reconstruct the differential equation of chaotic system under different noise levels; then, the reconstructed equation is embedded as a physical constraint into the loss function of physical information neural network (PINN) to achieve high-precision prediction of the future state of the system.
[0009] To achieve the above objectives, the present invention adopts the following technical solution:
[0010] This invention provides a method for modeling and predicting chaotic systems based on graph neural network symbolic regression algorithm, including the following steps:
[0011] S1: Construct a symbolic expression tree graph structure, represent the differential equation to be discovered as a binary tree structure, and perform graph processing on the tree structure to form a symbolic expression tree graph with self-loops;
[0012] S2: Based on a multi-layer graph neural network, feature propagation is performed on the symbol expression tree graph, and the symbol type is predicted node by node according to the preorder traversal order to generate a valid mathematical expression;
[0013] S3: Introduce graph neural network optimization strategies, including exploration reward and similarity-weighted policy gradients, to train and optimize the sampling process;
[0014] S4: The reconstructed differential equations are used as prior physical knowledge and embedded into the loss function of the physical information neural network to achieve modeling and prediction of the state of chaotic systems.
[0015] In one embodiment, in step S1:
[0016] The binary tree is a fixed-depth complete binary tree with a depth of 5 and a total of 31 nodes;
[0017] Each node corresponds to a mathematical symbol, including operators, system state variables, or optimizable numerical constants;
[0018] The graph processing includes establishing directed edges between each parent node and its left and right child nodes, and adding self-loop edges to each node to form a symbolic expression tree graph with self-loops.
[0019] The adjacency matrix of the symbolic expression tree graph is then subjected to symmetric normalization.
[0020] In one embodiment, step S2 specifically includes:
[0021] Initialize a feature matrix, where each row corresponds to the current state of a node;
[0022] In the i-th step, message passing is performed on the current graph structure based on a multi-layer graph neural network to obtain the context representation of the i-th node, and the symbol probability distribution is calculated through an output layer:
[0023]
[0024] Among them, W o h is a trainable weight matrix. i This represents the context of the i-th node;
[0025] Structural and mathematical constraints are applied during the sampling process;
[0026] After sampling, the one-hot encoding of the symbol is written into the i-th row of the feature matrix X, and the next node is processed until the entire tree is completely filled; ensure that the generated expression is syntactically valid, mathematically valid, and strictly aligned with the hierarchical structure of the expression tree.
[0027] In one embodiment, the structural constraints include: leaf nodes can only sample variables or constants; the right child node of a unary operator must be empty;
[0028] The mathematical constraints include: prohibiting the generation of semantically invalid or redundant expressions.
[0029] In one embodiment, step S3 specifically includes:
[0030] An exploration reward is introduced into the graph neural network optimization strategy, which encodes the preorder traversal sequence of the generated expression into an embedding vector to encourage expressions with novel structures.
[0031] By introducing a similarity-weighted policy gradient from the graph neural network optimization strategy and using genetic programming as an external evaluator, the longest common subsequence similarity between the generated expression and the reference solution is calculated.
[0032] Using the similarity as the weight for policy gradient update, the graph convolutional network parameters are updated using the similarity-weighted PPO algorithm.
[0033] In one embodiment, the longest common subsequence similarity between the generated expression and the reference solution is calculated using the following formula:
[0034]
[0035] Where τ is the generation expression for each layer of the graph neural network; τ * The optimal expression in the current batch is evolved and optimized to obtain a high-quality reference solution; LCS(·) is the longest common subsequence function.
[0036] In one embodiment, in step S4, the physical information neural network is a fully connected neural network, with time t as input and system state u as output. θ (t);
[0037] The loss function includes a data fitting term and a physical constraint term, and takes the following form:
[0038]
[0039] Where λ1, λ2, λ3 are hyperparameters, u i Let x be the actual system state composed of x, y, and z. θ ,y θ ,z θ These are the predicted values from the neural network; These are the first derivatives of the corresponding neural network predictions; N is the total number of points in the data point cloud.
[0040] In one embodiment, the physical information neural network has a 4-layer fully connected network with 64 neurons per layer and an activation function of tanh; it is trained using the Adam optimizer with a learning rate of 0.001.
[0041] As can be seen from the above technical solution, compared with the prior art, the present invention has the following technical advantages:
[0042] (1) Structure alignment: Graph neural networks directly model the hierarchical structure of expression trees, avoiding the structural misalignment problem of sequence models such as RNNs;
[0043] (2) Noise robustness: It can still accurately recover the chaotic system equations under 5% Gaussian white noise;
[0044] (3) End-to-end prediction: realize the complete closed loop of "equation discovery → physical constraints → state prediction". Attached Figure Description
[0045] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.
[0046] Figure 1 This is a flowchart of the chaotic system modeling and prediction method based on graph neural network symbolic regression algorithm provided in this embodiment of the invention;
[0047] Figure 2 A complete schematic diagram of the chaotic system modeling provided in the embodiments of the present invention;
[0048] Figure 3 This is a block diagram of the symbolic regression model provided in the embodiments of the present invention. Detailed Implementation
[0049] 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.
[0050] This invention discloses a method for modeling and predicting chaotic systems based on a graph neural network symbolic regression algorithm, referring to... Figure 1 As shown, it includes the following steps:
[0051] S1: Construct a symbolic expression tree graph structure, represent the differential equation to be discovered as a binary tree structure, and perform graph processing on the tree structure to form a symbolic expression tree graph with self-loops;
[0052] S2: Based on a multi-layer graph neural network, feature propagation is performed on the symbol expression tree graph, and the symbol type is predicted node by node according to the preorder traversal order to generate a valid mathematical expression;
[0053] S3: Introduce graph neural network optimization strategies, including exploration reward and similarity-weighted policy gradients, to train and optimize the sampling process of the graph neural network;
[0054] S4: The reconstructed differential equations are used as prior physical knowledge and embedded into the loss function of the physical information neural network to achieve modeling and prediction of the state of chaotic systems.
[0055] The specific principle and process of this invention are as follows: Figure 2As shown, the process includes the following steps: constructing a symbolic expression tree structure, expression sampling based on a graph neural network (GCN), introducing exploration rewards and similarity-weighted policy gradient training, and constructing a physical information neural network for state prediction. Steps S1 to S4 are described in detail below.
[0056] In step S1, the differential equation to be discovered is represented as a complete binary tree of fixed depth, for example, depth 5, with 31 nodes. Each node corresponds to a mathematical symbol, including operators (such as +, -, ×, ÷), system state variables (such as x, y, z), or optimizable numerical constants, denoted as const. To facilitate message passing in the graph neural network, the tree structure is graph-based: directed edges are established between each parent node and its left and right child nodes, and an additional self-loop edge is added to each node, forming the so-called "Symbolic Expression Tree with Self-Loops" (SETSL). The adjacency matrix of this graph structure is then symmetrically normalized.
[0057] Defined as:
[0058] Where A is the adjacency matrix of the original tree structure, and I is the identity matrix (representing a self-loop). This is the degree matrix of A+I. The core of this step is to explicitly encode the hierarchical structure of the expression into a graph topology, providing a foundation for subsequent GCN modeling.
[0059] In step S2, a multi-layer graph neural network (GCN) is used to propagate features of the SETSL graph, predicting the symbol type node by node in the preorder traversal order (root → left → right). Specifically, a feature matrix is initialized. Each row corresponds to the current state of a node (initially set to 0). <empty>(Placeholder), L = {+, -, ×, ÷, x, y, z, const} is the symbol library. X represents a matrix; R represents the real number field; V represents the number of nodes; |L| represents the size of the symbol library.
[0060] In the i-th step, GCN performs message passing on the current graph structure to obtain the context representation h of the i-th node. i And calculate the symbol probability distribution through an output layer:
[0061]
[0062] Among them, W o This is a trainable weight matrix. Subsequently, two types of hard constraints are applied to this distribution before sampling:
[0063] (1) Structural constraints: Leaf nodes are those with no child nodes and can only sample variables or constants; the right child node of a unary operator must be empty, such as sin.
[0064] (2) Mathematical constraints: Prohibit the generation of semantically invalid or redundant expressions, such as nested trigonometric functions sin(cos(x)) and combinations of logarithms and exponents that are inverse functions of each other, such as exp(log(x)).
[0065] After sampling, the one-hot encoding of the symbol is written to the i-th row of the feature matrix X, and the process continues to the next node until the entire tree is completely filled. This process ensures that the generated expression is syntactically valid, mathematically sound, and strictly aligned with the hierarchical structure of the expression tree.
[0066] In step S3, a dual mechanism is designed to optimize the sampling strategy of GCN, such as... Figure 3 As shown, this prevents premature convergence and improves exploration efficiency. Figure 3 In the process, the θ-driven GCN model generates batches of expressions τ1, τ2, τ3. 3...... τ n The exploration reward is encoded and a distribution is constructed to calculate the reward Rc for each expression; the similarity weighting module calculates similarity scores ω1, ω2, ω3 for each expression based on the optimization results of genetic programming GP and the set of expressions τ; both are involved in the training of parameter θ.
[0067] (1) Exploration Bonus: For each batch, e.g., 1000 expressions, the generated expressions are first encoded into a fixed-dimensional embedding vector based on their preorder traversal sequence. Specifically, for each symbol a... t Using one-hot encoding e(a) t This is combined with positional encoding for modulation, thereby encouraging the model to generate candidate expressions with novel structures;
[0068] (2) Similarity-weighted strategy gradient (SWPG): Genetic programming GP is invoked with a population size of 1100 and a generation count of 30 as an external evaluator to perform evolutionary optimization on the best expression in the current batch, thereby obtaining a high-quality reference solution τ. * Subsequently, the generation expressions τ and τ for each GCN are calculated. * LCS similarity between the longest common subsequences:
[0069]
[0070] LCS(·) is the longest common subsequence function;
[0071] This is then used as the weight for policy gradient updates. Crucially, the GP only provides reward signals and does not serve as training samples, thus avoiding the gradient explosion problem caused by distribution shifts in traditional hybrid methods. Finally, a similarity-weighted proximal policy optimization algorithm is used to update the GCN parameters, achieving a balance between exploration and exploitation.
[0072] In step S4, the differential equations successfully reduced in step S3, for example... As prior physical knowledge, the loss function of the PINN neural network embedding physical information.
[0073] Specifically, a fully connected neural network is constructed, with time t as input and system state u as output. θ (t)=(x θ (t),y θ (t),z θ (t)). The loss function consists of two parts:
[0074] (1) Data fitting term: measures the mean square error between the network output and the noisy observation data;
[0075] (2) Physical constraint terms: The derivative of the network output with respect to time is calculated through automatic differentiation, such as... And force it to conform to the reduction equation f x (x θ ,y θ The output should be as consistent as possible.
[0076] The loss function includes a data fitting term and a physical constraint term, and the total loss is in the form of:
[0077]
[0078] Where λ1, λ2, λ3 are hyperparameters, u i Let x be the actual system state composed of x, y, and z. θ ,y θ ,z θ These are the predicted values from the neural network; , , are the first derivatives of the corresponding neural network predictions; N is the total number of points in the data point cloud. After training, this PINN can not only fit the data within the observation time range, but also predict future states, even if the training data only covers a portion of the trajectory.
[0079] The chaotic system modeling and prediction method based on graph neural network symbolic regression algorithm provided by this invention can be deployed on computing platforms. As a joint symbolic regression and prediction algorithm, it aims to help relevant researchers to automatically model and predict chaotic system datasets, such as turbulence modeling in atmospheric flow and electron interference entanglement phenomena.
[0080] The implementation of this invention involves the following steps:
[0081] Step 1: This method uses the PyTorch framework to write GCN sampling, computation, and training code. PyTorch is a widely recognized deep learning framework in the industry. Its dynamic computation graph mechanism facilitates the implementation of complex graph neural network structures and automatic differentiation operations, providing a flexible and efficient implementation foundation for this invention.
[0082] Step 2: Construct the symbolic expression tree graph structure. Represent the target expression as a complete binary tree of depth 5 (31 nodes in total), with the symbol library L = {+,-,×,÷,x,y,z,const}. Construct the SETSL graph structure by introducing self-loop edges, and use a symmetric normalized adjacency matrix for message passing.
[0083] Step 3: Implement the GCN-Based Sampling module. A 4-layer GCN with a hidden dimension of 32 is used. Symbols are sampled node-by-node in preorder traversal, while structural and mathematical constraints are applied to ensure the validity of the generated expressions.
[0084] Step 4: Implement the exploration reward and similarity-weighted policy gradient module. The exploration reward is based on Gaussian density estimation; the SWPG module calls genetic programming, for example, with a population size of 1100 and a generation number of 30, to calculate the LCS similarity as the policy gradient weight; the similarity-weighted PPO algorithm is used to update the GCN parameters.
[0085] Step 5: Equation discovery under multi-noise levels. Using the Lorentz system as a typical example of a chaotic system, its trajectory data is generated and 0.1%, 1%, and 5% Gaussian white noise are added respectively. By modeling each equation separately, three differential equations were finally derived.
[0086] Step 6: Construct a physical information neural network and perform predictions. The network structure consists of 4 fully connected layers, 64 neurons per layer, tanh activation, with input time t and output (x, y, z). The loss function includes a data fitting term and three physical constraint terms, each with a weight of 1. The Adam optimizer is used for training with a learning rate of 0.001. Predictions are performed outside the training time range, such as when t ∈ [10, 15].
[0087] The results indicate that:
[0088] This invention conducts experiments on several typical chaotic system datasets, focusing on the Lorentz system as a representative example to verify the effectiveness of the proposed method in mining differential equations and constructing a physical information neural network for state prediction under different noise levels. The chaotic system differential equations used in the experiments are shown in Table 1, the expressions obtained by the proposed method are shown in Table 2, the efficiency comparison with the benchmark method is shown in Table 3, and the prediction performance of the physical information neural network constructed based on the reduced equations is shown in Table 4.
[0089] Table 1 presents the formulaic expressions for the differential equations of the chaotic system. This table lists the true differential equations corresponding to the three state variables of the Lorentz system, serving as a standard reference for subsequent symbolic regression tasks to evaluate the structural correctness and coefficient accuracy of the expressions mined in this invention.
[0090] Table 1. Formulaic Expression of Differential Equations for Chaotic Systems
[0091]
[0092] Table 2. Expressions obtained by this method under different noise levels.
[0093]
[0094] Note: All the restored expressions are structurally identical to the true equations, with only minor deviations in the constant coefficients, which meets the evaluation criterion of "prioritizing structural correctness" in symbolic regression.
[0095] Table 2 shows the expressions mined by this method under different noise levels. As can be seen from the table, under the three Gaussian white noise conditions of 0.1%, 1%, and 5%, the present invention can accurately recover expressions that are completely consistent with the actual equation structure, with only minor deviations in the constant coefficients, indicating that the present method has excellent noise robustness and structure discovery capability.
[0096] Table 3 Comparison of the number of sampling expressions required for the present invention and the benchmark method (Discover) to obtain the best results (unit: 10,000).
[0097] noise level method Average sampling size 5％ Discover 28.21 This invention 12.07
[0098] Table 3 compares the number of sampled expressions required for the present invention and the benchmark method (Discover) to obtain the best results (unit: 10,000). Discover is a representative method that first applied deep symbolic regression to the discovery of differential equations. Experimental results show that, under 5% high noise conditions, the present invention only requires 120,700 samples to reconstruct the three equations, while Discover requires an average of 282,100 samples, fully demonstrating the significant advantages of the present invention in search efficiency and stability.
[0099] Table 4 shows the state prediction accuracy (MSE) in the prediction interval t∈[10,15] after constructing the constraint terms of the Physical Information Neural Network (PINN) based on the differential equations restored in Table 2.
[0100] noise level Average MSE 0.1％ <![CDATA[1.43×10 -4 ]]> 1％ <![CDATA[4.27×10 -4 ]]> 5％ <![CDATA[1.37×10 -2 ]]>
[0101] Table 4 shows the state prediction accuracy (MSE) in the prediction interval t∈[10,15] after constructing the constraint terms of the Physical Information Neural Network (PINN) based on the differential equations restored in Table 2. The training data only covers t∈[0,10]. The results show that even under 5% high noise, PINN can still maintain good prediction ability, verifying the practicality of the present invention in end-to-end modeling and prediction tasks.
[0102] In summary, this invention proposes a chaotic system modeling and prediction method based on graph neural network symbolic regression algorithm, which can, to some extent, compensate for the shortcomings of existing physical information neural networks in fitting data with unknown control equations. Experimental results demonstrate that, compared with similar algorithms, this method has significant advantages in structure reconstruction success rate, noise robustness, sampling efficiency, and prediction accuracy.
[0103] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.
[0104] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.< / empty>
Claims
1. A method for modeling and predicting chaotic systems based on graph neural network symbolic regression algorithm, characterized in that, Includes the following steps: S1: Construct a symbolic expression tree graph structure, represent the differential equation to be discovered as a binary tree structure, and perform graph processing on the tree structure to form a symbolic expression tree graph with self-loops; S2: Based on a multi-layer graph neural network, feature propagation is performed on the symbol expression tree graph, and the symbol type is predicted node by node according to the preorder traversal order to generate a valid mathematical expression; S3: Introduce graph neural network optimization strategies, including exploration reward and similarity-weighted policy gradients, to train and optimize the sampling process; S4: The reconstructed differential equations are used as prior physical knowledge and embedded into the loss function of the physical information neural network to achieve modeling and prediction of the state of chaotic systems.
2. The method as described in claim 1, characterized in that, In step S1: The binary tree is a fixed-depth complete binary tree with a depth of 5 and a total of 31 nodes; Each node corresponds to a mathematical symbol, including operators, system state variables, or optimizable numerical constants; The graph processing includes establishing directed edges between each parent node and its left and right child nodes, and adding self-loop edges to each node to form a symbolic expression tree graph with self-loops. The adjacency matrix of the symbolic expression tree graph is then subjected to symmetric normalization.
3. The method as described in claim 1, characterized in that, Step S2 specifically includes: Initialize a feature matrix, where each row corresponds to the current state of a node; In the i-th step, message passing is performed on the current graph structure based on a multi-layer graph neural network to obtain the context representation of the i-th node, and the symbol probability distribution is calculated through an output layer: Among them, W o h is a trainable weight matrix. i This represents the context of the i-th node; Structural and mathematical constraints are applied during the sampling process; After sampling, the one-hot encoding of the symbol is written into the i-th row of the feature matrix X, and the next node is processed until the entire tree is completely filled; ensure that the generated expression is syntactically valid, mathematically valid, and strictly aligned with the hierarchical structure of the expression tree.
4. The method as described in claim 3, characterized in that, The structural constraints include: leaf nodes can only sample variables or constants; the right child node of a unary operator must be empty; The mathematical constraints include: prohibiting the generation of semantically invalid or redundant expressions.
5. The method as described in claim 1, characterized in that, Step S3 specifically includes: An exploration reward is introduced into the graph neural network optimization strategy, which encodes the preorder traversal sequence of the generated expression into an embedding vector to encourage expressions with novel structures. By introducing a similarity-weighted policy gradient from the graph neural network optimization strategy and using genetic programming as an external evaluator, the longest common subsequence similarity between the generated expression and the reference solution is calculated. Using the similarity as the weight for policy gradient update, the graph convolutional network parameters are updated using the similarity-weighted PPO algorithm.
6. The method as described in claim 5, characterized in that, The longest common subsequence similarity between the generated expression and the reference solution is calculated using the following formula: Where τ is the generation expression for each layer of the graph neural network; τ * The optimal expression in the current batch is evolved and optimized to obtain a high-quality reference solution; LCS(·) is the longest common subsequence function.
7. The method as described in claim 1, characterized in that, In step S4, the physical information neural network is a fully connected neural network, with time t as input and system state u as output. θ (t); The loss function includes a data fitting term and a physical constraint term, and takes the following form: Where λ1, λ2, λ3 are hyperparameters, u i Let x be the actual system state composed of x, y, and z. θ ,y θ ,z θ These are the predicted values from the neural network; These are the first derivatives of the corresponding neural network predictions; N is the total number of points in the data point cloud.
8. The method as described in claim 7, characterized in that, The physical information neural network has a 4-layer fully connected network structure with 64 neurons in each layer and an activation function of tanh. It is trained using the Adam optimizer with a learning rate of 0.001.