Mechanism information analysis method of nonlinear dynamic control system based on causal diagram

Abstract

The invention relates to a mechanism information analysis method of a nonlinear control system, which combines a space-time diagram convolutional network with mechanism information modeling, and learns a control equation of the system from observation data with noise. Firstly, causal graph coding based on transfer entropy is designed to obtain an adjacent matrix with causal significance between variables. Then, a space-time diagram convolutional network is used to obtain an approximate solution of a system variable. On this basis, automatic differentiation is applied to obtain a basic derivative, a dictionary of candidate algebraic terms is formed, and sparse regression is used to obtain a coefficient matrix to determine an explicit formula of the control equation. The invention also designs a new cross combination optimization strategy to learn heterogeneous parameters including neural network parameters and control equation coefficients. According to the method, the advantages of causal representation, deep learning, automatic differentiation and sparse regression are fused aiming at the limitation of numerical differentiation and redundancy calculation of an existing control equation learning method, the robustness and universality are high, and theoretical and practical significance is achieved for mechanism information discovery of a nonlinear dynamic system.

Description

Technical Field

[0001] The invention belongs to the technical field of mechanism information mining of nonlinear dynamic systems, in particular to a mechanism information analysis method of a nonlinear dynamic control system based on a cause-effect diagram. Background Art

[0002] In the field of practical engineering control, control equations are crucial. For example, how to accurately identify the aircraft dynamics equations, so as to design a controller based on the explicit control equations to achieve real-time control of the aircraft's flight angle (angle of attack). Another example is the control of industrial processes such as grinding, which also requires accurate description of the macro and micro mechanisms of the grinding process, accurate identification of the control equations, and design of controllers to control process variables such as the feed water flow rate in real time.

[0003] Modern machine learning methods have made significant progress in the black-box prediction performance of dynamic systems, but the simplified closed form of the system's internal control equations is still unclear or partially unknown. Therefore, the control task is completed under the premise of unknown internal mechanisms, and the ideal control accuracy cannot be achieved. The control operation is not mechanistically interpretable. Therefore, it is urgent to study how to explore system control equations containing potential mechanism information from observation data. In recent years, with the development of intelligent technology and the improvement of computing power, data-driven system control equation learning methods have developed rapidly. Research based on the symbolic regression framework has high computational cost and may have overfitting problems. Research based on the sparse regression framework also has two major drawbacks. First, the existing methods use ordinary numerical methods to calculate derivatives to construct control equations, such as finite differences or filtering. In the absence of data and high noise, the errors caused by approximate derivatives will reduce the accuracy of sparse regression-based methods. Another limitation is that a complete candidate function library will lead to information redundancy and computational complexity. Therefore, a mechanism information analysis method that can solve the above problems at the same time has important research significance. Summary of the invention

[0004] The present invention provides a mechanism information analysis method for a nonlinear dynamic control system based on a causal graph, which promotes the sparse learning of the interaction dynamic coefficients in the control equation by obtaining an adjacency matrix representing the causal relationship between the observed variables through a graph encoding method based on transfer entropy. At the same time, it can optimize the performance of spatiotemporal graph convolution. On this basis, the spatiotemporal graph convolution can capture the complex local spatiotemporal correlation of the variables and obtain an accurate approximation of the system solution. In the parameter optimization stage, the neural network provides an approximate solution of the system and accurate modeling of the derivatives for sparse regression, and then constrains and optimizes the parameter learning of the network from the control equation obtained from the sparse regression. At the same time, the framework meets the interpretability requirements, and the learned control equation can naturally become an explanation of the graph encoding.

[0005] The technical solution adopted by the present invention to achieve the above-mentioned purpose is:

[0006] A mechanism information analysis method for nonlinear dynamic control systems based on causal graphs, which combines spatiotemporal graph convolutional networks with mechanism information modeling to learn the control equations of the system from noisy observation data, includes the following steps:

[0007] Step 1: Obtain system observation variables;

[0008] Step 2: Encode the causal graph between variables based on transfer entropy to obtain the bidirectional causal correlation between variables;

[0009] Step 3: Combined with the causal graph adjacency matrix, the generated data is used to train the spatiotemporal graph convolutional network STGCN, which captures the complex temporal and spatial dependencies between variables and obtains the potential solution of the system.

[0010] Step 4: Use automatic differentiation to obtain an accurate calculation of the potential solution;

[0011] Step 5: Use the sparse regression module to determine the coefficients of each item in the candidate function library and obtain the display expression of the control equation;

[0012] Step 6: Use a cross-combination optimization strategy to alternately optimize the parameters of the spatiotemporal graph convolutional network and the coefficients of the control equation until convergence, and finally obtain the control equation that best matches the system.

[0013] The method further comprises: realizing accurate prediction of the system according to the obtained control equation, and quantitatively analyzing the state change of the system according to the explicit equation to realize precise control of the system.

[0014] The system observation variables are generated by generating noisy data in different sampling time periods according to the real control equation of the system.

[0015] The causal graph encoding comprises the following steps:

[0016] The transfer entropy matrix is ​​used as the adjacency matrix A to describe the causal interactions between the observed variables. The element A ij Represents x i With x j The causal relationship between them, when i = j, A ij =1; when i≠j, A ij The calculation is as follows:

[0017]

[0018] Among them, causal relationship when When x i is x j Cause; Then it means x i is x jThe result of transferring entropy between variables is based on the assumption that the system dynamics is Markov, P(x i,t ,x i,t-1 ,x j,t-1 ) is the joint probability density function, P(x i,t |x i,t-1 ,x j,t-1 ) and P(x i,t |x i,t-1 ) is the conditional probability density function.

[0019] The spatiotemporal graph convolutional network includes: k spatiotemporal layers STGCN and two linear layers stacked in sequence;

[0020] Each spatiotemporal layer STGCN extracts temporal dependencies of different frequencies through expansion factors of different sizes, and performs a residual connection between the output of the current graph convolutional network GCN layer and the initial input through the linear layer;

[0021] The two linear layers are used to output a sequence: the potential solutions to the governing equations.

[0022] Each of the spatiotemporal layers comprises the following steps:

[0023] Step 3-1: The input variable first passes through a linear layer and then passes through a gated temporal convolutional network (Gated TCN) consisting of two parallel dilated causal convolutions to capture the dynamic trends of complex observed variables. TCN is constructed by combining p convolutional filters W i (i=1,2,…p) with different sizes of convolution kernels 1×k i (i=1,2,…p) are connected in series to extract the time features of different frequencies; the process is expressed as follows:

[0024] H = tanh(h 1 )⊙σ(h 2 )

[0025]

[0026]

[0027]

[0028] Where tanh(·) is the hyperbolic tangent activation function, σ(·) is the sigmoid activation function, * is the convolution operation, ⊙ represents the Hadamard product of the elements, and concat(·) is the concatenation operation. The parameter d is the dilation factor;

[0029] Step 3-2: Perform graph convolution (GCN) operation to aggregate node information and neighborhood information, which is as follows:

[0030]

[0031] Where A is the adjacency matrix and D is a degree matrix. ii =∑ j A ij , H is the hidden state output by the gated TCA module, and W is the trainable model parameter matrix;

[0032] Step 3-3: The output of GCN is residually connected to the initial input.

[0033] The sparse regression module includes:

[0034] Step 5-1 Split the control equation into self-dynamics and interaction dynamics:

[0035]

[0036] F(x i (t)) is a linear or nonlinear function describing the autodynamics of all observed variables, G(x i (t),x j (t)) aims to capture the dynamic mechanism of interaction between variables; further:

[0037]

[0038] Where A is the adjacency matrix obtained in step 2-3, Φ F is a dictionary containing function entries describing the autodynamics, Φ G is a library of candidate functions for describing interaction dynamics; Φ F (x i (t)) and Φ G (x i (t),x j (t)) is a time-varying matrix; the symbol represents the Kronecker product, ξ F and G is the sparse coefficient to be inferred;

[0039] Step 5-2: Construct a candidate function library Φ based on the exact derivative obtained in step 4 F and Φ G , including derivatives of various orders, as well as polynomials, trigonometric functions, exponential functions, and common activation functions;

[0040] Step 5-3: Obtain the coefficient matrix ξ of the control equation through sparse regression F and G .

[0041] The cross-combination optimization strategy specifically includes:

[0042] Step 6-1. According to the variables that need to be optimized simultaneously in the network architecture: STGCN parameters and coefficients of the control equation, the total loss function of the training is defined as:

[0043]

[0044] Where W is the trainable parameter matrix of STGCN, X is the set of input observation variables, α and β are the relative weights of the loss function, is the data reconstruction loss of STGCN, is the physical loss for learning physical information, Yes 0 Regularization term, used to improve the coefficient matrix ξ F and G The sparsity of ; the loss function is specifically defined as:

[0045]

[0046]

[0047]

[0048] Where N is the number of input observation variables, is the reconstructed state obtained by STGCN, ||·|| 0 and ||·|| 2 denote the L0 norm and the Frobenius norm respectively.

[0049] Step 6-2: Define the cross-combination optimization strategy, decompose the optimization of the total loss, and implement heterogeneous parameters W through alternating iterations k+1 , {ξ F ,ξ G} k+1 Optimization:

[0050]

[0051]

[0052] Step 6-3. In each iteration, use the Adam optimizer to train the parameters of STGCN, set the initial learning rate, stack several spatiotemporal graph convolution layers, and update the sparse differential equation coefficients of the first k through STRidge based on the k-1th iteration result of the STGCN parameters, and loop until the heterogeneous parameters converge.

[0053] The present invention has the following beneficial effects and advantages:

[0054] A new causal graph-based mechanism information analysis method for nonlinear dynamic systems is used to discover the potential control equations of nonlinear dynamic systems from noisy multidimensional observation data. The method combines transfer entropy-based causal graph embedding to mine causal relationships, spatiotemporal graph convolutional networks for nonlinear function approximation, automatic differentiation to obtain exact derivatives, and sparse regression to obtain coefficient matrices. The adjacency matrix containing the causal correlations of variables obtained by encoding the causal graph based on transfer entropy can promote sparse learning of interaction dynamics coefficients and accelerate convergence. The convolution of the spatiotemporal graph obtains the potential state of the system and is connected to the physical information sparse regression through automatic differentiation as the basis of sparse regression. The control equations obtained by sparse regression in turn constrain and promote the parameter optimization of the spatiotemporal graph convolutional network. Therefore, the global optimization adopts a cross-combination optimization algorithm to obtain the coefficient matrix of the control equation that best matches the system while optimizing the neural network parameters. The mechanism information equations discovered from the data can accurately guide system predictive control. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 The present invention is a flow chart of the method.

[0056] Figure 2 A comparison diagram of the full-field waves and space-time responses generated by the true KS equation and the equation discovered by the present invention under two initial conditions.

[0057] Figure 3 Diagram of the spatiotemporal graph convolutional network structure designed for the present invention.

[0058] Figure 4 It is a comparison diagram of the full-field waves generated by the real Fitzhugh-Nagumo equation and the control equation of the present invention under three initial conditions.

[0059] Figure 5 It is a comparison diagram of the real adjacency matrix and the adjacency matrix discovered by the present invention.

[0060] Figure 6 For real Comparison chart of dynamic response trajectories generated by the equation and the equation discovered by the present invention. DETAILED DESCRIPTION

[0061] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the specific implementation method of the present invention is described in detail below in conjunction with the accompanying drawings. In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without violating the connotation of the invention, so the present invention is not limited by the specific implementation disclosed below.

[0062] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art of the present invention. The terms used in the specification of the invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention.

[0063] The present invention is a unified framework for discovering system mechanism information based on the control requirements of industrial systems. The present invention attempts to verify several typical dynamic systems, including the KS equation, the Fitzhugh-Nagumo equation and The experimental results show that the present invention can discover the underlying mechanism information of the system and has strong robustness and universality. Therefore, the present invention can be extended to general industrial systems, discover the inherent control equations of the system, and guide actual industrial control based on the explicit control relationship.

[0064] like Figure 1 As shown, it is a flow chart of an example method of the present invention. A method for analyzing mechanism information of a nonlinear dynamic system based on a causal graph first needs to obtain system observation data. The present invention generates noisy data based on the real control equation of a typical system, and designs a causal graph encoding based on transfer entropy to obtain an adjacency matrix with causal significance between variables. Then, a spatiotemporal graph convolutional network is used to obtain an approximate solution for the system variables. On this basis, automatic differentiation is applied to obtain basic derivatives, a dictionary of candidate algebraic terms is formed, and sparse regression is used to obtain a coefficient matrix to determine the explicit formula of the control equation from the observation data. The present invention designs a new cross-combination optimization strategy to learn heterogeneous parameters including neural network parameters and control equation coefficients. The programming language used in the program execution step of the present invention is not limited to MATLAB, Python, etc.

[0065] The specific steps of the present invention are as follows:

[0066] Step 1: Obtain system observation data.

[0067] Example: Generate noisy data from the real governing equations of a typical system: The equation generates a test data set in the time period [0, T] = [0, 100], where dt = 0.01. At the same time, in order to verify the robustness of the algorithm, 10% noise is added to the data. The adjacency matrix with an average total degree of 5.1 is randomly generated.

[0068] True equation:

[0069]

[0070] Among them, u i 、v i、w i represents the state of the system, ε = 0.1 is the coupling strength, A ij is the adjacency matrix element.

[0071] Step 2: Encode the causal graph between variables based on transfer entropy to obtain the bidirectional causal correlation between variables;

[0072] Step 2-1. Calculate variable x i With x j The transfer entropy between:

[0073]

[0074] where x i,t is the variable x i The value at time t, P(x i,t ,x i,t-1 ,x j,t-1 ) is the joint probability density function, P(x i,t |x i,t-1 ,x j,t-1 ) and P(x i,t |x i,t-1 ) is the conditional probability density function;

[0075] Step 2-2: Calculate the causal relationship between variables based on transfer entropy:

[0076]

[0077] when When x i is x j Cause; Then it means x i is x j Fruit.

[0078] Step 2-3: Use the calculated transfer entropy as the prior information of the system, that is, use the transfer entropy matrix as the adjacency matrix A to describe the causal interaction between the observed variables. The element A ij Represents x i With x j The causal relationship between them, when i = j, A ij =1; when i≠j, A ij The calculation is as follows:

[0079]

[0080] Example: For real homogeneous The noisy data generated by the equation is encoded into a causal graph, and the comparison between the obtained adjacency matrix A and the true adjacency matrix is ​​as follows Figure 2 .

[0081] Step 3: Combined with the adjacency matrix of the causal graph, the generated data (noisy variables u, v, w) is used to train the spatiotemporal graph convolutional network (including k spatiotemporal layers and linear layers stacked in sequence; the output of each spatiotemporal layer is the output of the current GCN layer and the initial input through the linear layer for residual connection;), while capturing the complex temporal and spatial dependencies between variables to obtain the potential solution of the system. The network structure diagram is shown in Figure 3 .

[0082] Step 3-1: The input variable first passes through a linear layer and then passes through a gated TCN containing two parallel dilated causal convolutions to capture the dynamic trends of complex observed variables. TCN is constructed by combining p convolutional filters W i (i=1,2,…p) with different sizes of convolution kernels 1×k i (i=1,2,…p) are connected in series to extract the time features of different frequencies. The process is expressed as follows:

[0083] H = tanh(h 1 )⊙σ(h 2 ) (5)

[0084]

[0085]

[0086]

[0087] Among them, x is defined as the noisy system observation variable, which is u, v, and w in this example, tanh(·) is the hyperbolic tangent activation function, σ(·) is the sigmoid activation function, * is the convolution operation, ⊙ represents the Hadamard product of the elements, and concat(·) is the concatenation operation. The parameter d is the expansion factor, and s represents the number of layers in which the convolution is located.

[0088] Step 3-2: After extracting the temporal features of the observed variables, the input observed variable matrix can be converted into a graph feature matrix by combining the adjacency matrix obtained by the graph learning module. The nodes in the graph represent the hidden state of each observed variable output by the temporal convolution module, and the adjacency matrix represents the causal relationship between the nodes. Perform graph convolution operations to aggregate node information and neighborhood information, and output the convolution result Z, which is in the following form:

[0089]

[0090] Where A is the adjacency matrix obtained by step 2-3, and D is a degree matrix, where D ii =∑ j A ij , H is the hidden state output by the gated TCA module, and W is the trainable model parameter matrix.

[0091] Step 3-3, the output of GCN is connected to the initial input residual to prevent information loss during training. By stacking k layers, STGCN is able to extract temporal dependencies of different frequencies through expansion factors of different sizes. On this basis, causal dependencies at different time levels can be processed. The output part of STGCN is to connect the results of k spatiotemporal layers, and then obtain the output sequence through two linear layers: the potential solution u of the control equation * ,v * ,w * .

[0092] Step 4: Use automatic differentiation to obtain u * ,v * ,w * The derivative of

[0093] Step 5: Use the sparse regression module to determine the coefficients of each item in the candidate function library and obtain the display expression of the control equation;

[0094] Step 5-1, split the general control equation into self-dynamics and interaction dynamics:

[0095]

[0096] F(x i (t)) is a linear or nonlinear function describing the autodynamics of all observed variables, G(x i (t),x j (t)) aims to capture the dynamic mechanism of the interaction between variables. Further:

[0097]

[0098] Where A is the adjacency matrix obtained in step 2-3, Φ F is a dictionary containing function entries describing the autodynamics, Φ G is a candidate library of functions for describing interaction dynamics. Φ F (x i (t)) and Φ G (x i (t),x j (t)) is a time-varying matrix. Symbol represents the Kronecker product, ξ F and G are the sparse coefficients to be inferred.

[0099] Step 5-2: Construct a candidate function library Φ based on the exact derivative obtained in step 4 F and Φ G, including derivatives of various orders, as well as polynomials, trigonometric functions, exponential functions, and common activation functions.

[0100] Step 5-3: Obtain the coefficient matrix ξ of the control equation through sparse regression F and G .

[0101] Step 6: Use a cross-combination optimization strategy to alternately optimize the parameters of the spatiotemporal graph convolutional network and the coefficients of the control equation until convergence, and finally obtain the control equation expression that best matches the system.

[0102] Step 6-1. The designed network architecture needs to optimize variables including STGCN parameters and control equation coefficients at the same time, so the total loss function of training is defined as:

[0103]

[0104] Where W is the trainable parameter matrix of STGCN, X is the set of input observation variables, α and β are the relative weights of the loss function, is the data reconstruction loss of STGCN, is the physical loss for learning physical information, Yes 0 Regularization term, used to improve the coefficient matrix ξ F and G These loss functions are specifically defined as:

[0105]

[0106]

[0107]

[0108] Where N is the number of input observation variables, is the reconstructed state obtained by STGCN, ||·|| 0 and ||·|| 2 denote the L0 norm and the Frobenius norm respectively. It is worth noting that Ensure that STGCN accurately approximates the potential solution of the system by fitting the original data and connects to the mechanism information discovery module through automatic differentiation, The closed form of the system control equations is reconstructed to provide constraints for STGCN, and Regularization ensures a sparse representation of the differential equation.

[0109] Step 6-2: It is very difficult to directly solve the optimization problem of the total loss function because l 0The non-convex optimization of is an NP-hard problem. In order to solve this problem, a cross-combination optimization strategy is designed to decompose the optimization of the total loss into a set of easy-to-solve sub-problems, and optimize heterogeneous parameters through alternating iterations. In other words:

[0110]

[0111]

[0112] Step 6-3, in this example, in each iteration, the parameters of STGCN are trained using the Adam optimizer with an initial learning rate of 0.001. Six spatiotemporal graph convolutional layers are stacked with expansion factors {1, 2, 1, 2, 1, 2} and 16 output channels. In each spatiotemporal graph convolutional layer, four filter sizes are used, i.e., p = 4, k in step 3-1. i ={2,3,6,7} to generate convolution kernels of size 1×2, 1×3, 1×6 and 1×7 to cover different time periods. And the jump connection layer has 32 output channels. Based on the k-1th iteration result of the STGCN parameters, the sparse differential equation coefficients of the first k are updated by STRidge and looped until convergence. Thus, the present invention finds The equation is as follows:

[0113]

[0114] like Figure 3 For real Comparison of dynamic response trajectories generated by the equation and the equation discovered by the present invention. In addition to this example, the present invention also tests the noisy data generated by the KS equation and the Fitzhugh-Nagumo equation. Figure 4 A comparison diagram of the full-field waves and space-time responses generated by the true KS equation and the equation discovered by the present invention under two initial conditions. Figure 5 The figure is a comparison of the full-field waves generated by the real Fitzhugh-Nagumo equation and the equation discovered by the present invention under three initial conditions. It can be seen that the mechanism information of the nonlinear dynamic system, that is, the underlying control equation, can be discovered from noisy data using this method.

[0115] Step 7: Accurately predict the system based on the obtained control equation, and quantitatively analyze the system state changes based on the explicit equation to achieve precise control of the system.

[0116] The present invention automatically discovers the underlying control equation of the system from the observed data, has universality, and can be generalized to the field of actual engineering control. For example, based on the flight trajectory data of the aircraft, the present invention is applied to discover the aircraft dynamics equation, and on the basis of the explicit control equation, a controller such as PID can be designed to control the flight angle (angle of attack) of the aircraft in real time. The present invention can also be applied to industrial process control such as grinding, that is, the present invention is applied to discover the macroscopic and microscopic mechanisms of the grinding process, and a controller is designed according to the control equation to control process variables such as the flow rate of additional water in real time.

[0117] In summary, the present invention effectively integrates scientific methods driven by data and physical information: the adjacency matrix containing the causal correlation of variables obtained by causal graph encoding based on transfer entropy can promote sparse learning of interaction dynamics coefficients and accelerate convergence. The convolution of the space-time graph obtains the potential state of the system and is connected to the physical information sparse regression through automatic differentiation as the basis of sparse regression. The control equations obtained by sparse regression in turn constrain and promote the parameter optimization of the space-time graph convolutional network. A cross-combination optimization algorithm is used to obtain the coefficient matrix of the control equation that best matches the system while optimizing the neural network parameters. It combines the advantages of causal representation, deep learning, automatic differentiation and sparse regression, and shows excellent effectiveness and robustness. The explicit system control relationship obtained by the present invention can guide actual industrial control.

[0118] The embodiments described in the above description will help those skilled in the art to further understand the present invention, but do not limit the present invention in any form. It should be noted that, for those of ordinary skill in the art, several changes and improvements can be made without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

Claims

1. A mechanism information analysis method for nonlinear dynamic control systems based on causal graphs, characterized in that: This method combines spatiotemporal graph convolutional networks with mechanism information modeling to learn the control equations of the system from noisy observation data. It includes the following steps: Step 1: Obtain system observation variables; Step 2: Encode the causal graph between variables based on transfer entropy to obtain the bidirectional causal correlation between variables; Step 3: Combined with the causal graph adjacency matrix, the generated data is used to train the spatiotemporal graph convolutional network STGCN, which captures the complex temporal and spatial dependencies between variables and obtains the potential solution of the system. Step 4: Use automatic differentiation to obtain an accurate calculation of the potential solution; Step 5: Use the sparse regression module to determine the coefficients of each item in the candidate function library and obtain the display expression of the control equation; Step 6: Use a cross-combination optimization strategy to alternately optimize the parameters of the spatiotemporal graph convolutional network and the coefficients of the control equation until convergence, and finally obtain the control equation that best matches the system.

2. The nonlinear system mechanism information analysis method according to claim 1, characterized in that: The method further comprises: realizing accurate prediction of the system according to the obtained control equation, and quantitatively analyzing the state change of the system according to the explicit equation to realize precise control of the system.

3. The nonlinear system mechanism information analysis method according to claim 1, characterized in that: The system observation variables are generated by generating noisy data in different sampling time periods according to the real control equation of the system.

4. The nonlinear system mechanism information analysis method according to claim 1, characterized in that: The causal graph encoding comprises the following steps: The transfer entropy matrix is ​​used as the adjacency matrix A to describe the causal interactions between the observed variables. The element A ij Represents x i With x j The causal relationship between them, when i = j, A ij =1; when i≠j, A ij The calculation is as follows: Among them, causal relationship when When x i is x j Cause; Then it means x i is x j The result of the transfer entropy TE between variables xi→xj is based on the assumption that the system dynamics is Markov, P(x i,t ,x i,t-1 ,x j,t-1 ) is the joint probability density function, P(x i,t |x i,t-1 ,x j,t-1 ) and P(x i,t |x i,t-1 ) is the conditional probability density function.

5. The nonlinear system mechanism information analysis method according to claim 1, characterized in that: The spatiotemporal graph convolutional network includes: k spatiotemporal layers STGCN and two linear layers stacked in sequence; Each spatiotemporal layer STGCN extracts temporal dependencies of different frequencies through expansion factors of different sizes, and performs a residual connection between the output of the current graph convolutional network GCN layer and the initial input through the linear layer; The two linear layers are used to output a sequence: the potential solutions to the governing equations.

6. The nonlinear system mechanism information analysis method according to claim 5, characterized in that: Each of the spatiotemporal layers comprises the following steps: Step 3-1: The input variable first passes through a linear layer and then passes through a gated temporal convolutional network (Gated TCN) consisting of two parallel dilated causal convolutions to capture the dynamic trends of complex observed variables. TCN is constructed by combining p convolutional filters W i (i=1,2,…p) with different sizes of convolution kernels 1×k i (i=1,2,…p) are connected in series to extract the time features of different frequencies; the process is expressed as follows: H=tanh(h1)⊙σ(h2) Where tanh(·) is the hyperbolic tangent activation function, σ(·) is the sigmoid activation function, * is the convolution operation, ⊙ represents the Hadamard product of the elements, and concat(·) is the concatenation operation. The parameter d is the dilation factor; Step 3-2: Perform graph convolution (GCN) operation to aggregate node information and neighborhood information, which is as follows: Where A is the adjacency matrix and D is a degree matrix. ii =∑ j A ij , H is the hidden state output by the gated TCA module, and W is the trainable model parameter matrix; Step 3-3: The output of GCN is residually connected to the initial input.

7. A nonlinear system mechanism information analysis method according to claim 1, characterized in that: The sparse regression module includes: Step 5-1 Split the control equation into self-dynamics and interaction dynamics: F(x i (t)) is a linear or nonlinear function describing the autodynamics of all observed variables, G(x i (t),x j (t)) aims to capture the dynamic mechanism of interaction between variables; further: Where A is the adjacency matrix obtained in step 2-3, Φ F is a dictionary containing function entries describing the autodynamics, Φ G is a library of candidate functions for describing interaction dynamics; Φ F (x i (t)) and Φ G (x i (t),x j (t)) is a time-varying matrix; the symbol represents the Kronecker product, ξ F and G is the sparse coefficient to be inferred; Step 5-2: Construct a candidate function library Φ based on the exact derivative obtained in step 4 F and Φ G , including derivatives of various orders, as well as polynomials, trigonometric functions, exponential functions, and common activation functions; Step 5-3: Obtain the coefficient matrix ξ of the control equation through sparse regression F and G .

8. A nonlinear system mechanism information analysis method according to claim 1, characterized in that: The cross-combination optimization strategy specifically includes: Step 6-1. According to the variables that need to be optimized simultaneously in the network architecture: STGCN parameters and coefficients of the control equation, the total loss function of the training is defined as: Where W is the trainable parameter matrix of STGCN, X is the set of input observation variables, α and β are the relative weights of the loss function, is the data reconstruction loss of STGCN, is the physical loss for learning physical information, is the l0 regularization term, used to improve the coefficient matrix ξ F and G The sparsity of ; the loss function is specifically defined as: Where N is the number of input observation variables, is the reconstructed state obtained by STGCN, ||·||0 and ||·||2 represent the L0 norm and Frobenius norm, respectively. Step 6-2: Define the cross-combination optimization strategy, decompose the optimization of the total loss, and implement heterogeneous parameters W through alternating iterations k+1 , {ξ F ,ξ G } k+1 Optimization: Step 6-3. In each iteration, use the Adam optimizer to train the parameters of STGCN, set the initial learning rate, stack several spatiotemporal graph convolution layers, and update the sparse differential equation coefficients of the first k through STRidge based on the k-1th iteration result of the STGCN parameters, and loop until the heterogeneous parameters converge.

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