A physics-based method for discovering governing equations from scarce and noisy data
By combining physical information neural networks and sparse regression methods, and utilizing deep neural network fitting and fine-tuning, the challenge of discovering complex system control equations in scarce and noisy data has been solved, achieving efficient and accurate identification and prediction of control equations.
Patent Information
- Application Number
- CN202510240623.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2025-03-03
- Publication Date
- 2025-11-18
- Estimated Expiration
- 2045-03-03
AI Technical Summary
Existing technologies face challenges such as high computational costs, overfitting, and sensitivity to noise when discovering the governing equations of complex systems from scarce and noisy data, making it difficult to effectively uncover physical mechanisms.
By combining physical information neural networks and sparse regression methods, the candidate function library is reduced through dimensional verification, and deep neural networks are used for fitting and automatic differentiation. Sparse regression is used to identify the control equations, and the coefficients are fine-tuned through deep neural networks.
It enables efficient and accurate discovery of the control equations of complex dynamic systems under conditions of scarce and noisy data, and has good robustness and versatility, making it suitable for engineering applications.
Smart Images
Figure CN120105365B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of mechanism information mining, modeling and prediction of complex dynamic systems, and in particular relates to a physics-based method for discovering governing equations from scarce and noisy data. Background Technology
[0002] Discovering the governing equations describing the dynamics of complex systems from data has long been a central challenge in applied mathematics and engineering. Traditionally, the derivation of governing partial differential equations (PDEs) has relied heavily on theoretical principles and empirical observations. However, with the advent of big data and advancements in machine learning (ML), there has been increasing interest in data-driven approaches to discovering governing equations over the past decade.
[0003] The key technique for discovering partial differential equations (PDEs) using data-driven methods is model regression, where all terms in the regression model are predetermined, and their corresponding coefficients are derived through machine learning algorithms. Common methods for discovering PDEs from data can be broadly categorized into two types: symbolic regression and sparse regression. Early symbolic regression methods often combined genetic programming to discover governing equations from low-dimensional system data. However, using genetic programming to encode equations through nonlinear analytic trees of varying sizes and structures can lead to complexity in high-dimensional scenarios, increasing computational costs and potential overfitting. Another regression method is sparse regression, with the widely used nonlinear dynamics sparse identification (SINDy) method being a prime example. This method utilizes sparse regression to select candidate functions from a high-dimensional nonlinear function space, effectively revealing simplified governing equations, especially ordinary differential equations. However, sparse regression is highly dependent on the quality and quantity of data, particularly numerical differentiation, which makes this method sensitive to noise. With the development of machine learning (ML) and deep learning (DL), automatic differentiation offers a viable solution to these problems, achieving success in both forward and backward analysis of nonlinear partial differential equations in Physical Information Neural Networks (PINN). However, the PINN framework for discovering physical laws faces some limitations, including high computational costs and the challenge of designing a predefined library of appropriately sized candidates. Summary of the Invention
[0004] This invention addresses the limitations of existing methods for mining mechanistic information in dynamic systems by proposing a physics-based method for discovering governing equations from scarce and noisy data.
[0005] This invention combines Physical Information Neural Network (PINN) with sparse regression to discover partial differential governing equations (PDEs) of dynamic systems from scarce and noisy data. First, a dimensional verification method effectively reduces the size of the candidate function library. Then, the powerful nonlinear fitting capability and automatic differentiation properties of deep neural networks (DNNs) are utilized to model the physical system and compute the candidate functions. Finally, sparse regression yields the form and coefficients of the governing equations, which are then fine-tuned using the DNN. This method not only discovers governing equations from data but also derives a network model for predicting the response of dynamic systems. The method is simple, efficient, accurate, and versatile. It can be widely applied to the mining, modeling, and reasoning of physical knowledge in complex dynamic systems. Its simplicity, high accuracy, and versatility make it particularly suitable for engineering applications. Compared to traditional methods, the proposed method exhibits better robustness and accuracy. Furthermore, this invention demonstrates good versatility and scalability.
[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:
[0007] A physics-based method for discovering governing equations from scarce and noisy data combines a physics-information neural network with a sparse regression method to discover the governing equations of a dynamic system from scarce and noisy data, comprising the following steps:
[0008] Step 1: Obtain the field data u and physical constants μ of the dynamic system through numerical simulation or experimental observation, and construct a measurement dataset by combining u with the corresponding time and space coordinate data.
[0009] Step 2: Obtain the residual point dataset by sampling using the Latin hypercube sampling method.
[0010] Step 3: Construct and initialize a deep neural network, which consists of fully connected neural networks, to fit the solution to the problem and compute the candidate library Φ. θ The function terms in the text.
[0011] Step 4: Train the deep neural network using the measurement dataset and residual point dataset to obtain a preliminary trained network model and equation coefficients.
[0012] Step 5: Use the ridge regression algorithm with sequential thresholds to identify the sparsity of the field variables of the dynamic system, and obtain the form and coefficients of the partial differential control equations of the dynamic system.
[0013] Step 6: Once the partial differential governing equations of the dynamic system are identified, the equation coefficients are fine-tuned using a deep neural network. Simultaneously, the trained neural network can also predict the system's state at any given time.
[0014] Furthermore, the candidate library Φ θ It was obtained by performing dimensional verification on the initial incomplete candidate library.
[0015] Furthermore, the initial overcomplete candidate library includes physical constant terms, polynomial terms, partial derivative terms, and combinations thereof. Performing dimensional verification on the initial overcomplete candidate library before network training not only ensures that the candidate library has practical physical meaning but also reduces its size, thereby improving computational efficiency.
[0016] Furthermore, it was found that the equation process includes the following sub-steps:
[0017] Step 4.1: Pre-train the deep neural network using the measurement dataset and residual point dataset, while optimizing the network parameters θ and equation coefficients λ.
[0018] Step 4.2: The pre-trained network outputs the time derivative vector of the target variable. The candidate library is calculated using automatic differentiation.
[0019] Step 4.3: Perform sparse regression on the system. Terms in the candidate library that are related to the system's governing equations are activated, thereby obtaining the form and coefficients of the equations.
[0020] Step 4.4 involves post-training using a deep neural network to fine-tune the obtained equation coefficients and further optimize the network parameters to improve fitting accuracy.
[0021] Furthermore, the optimization objective of the pre-training process is:
[0022]
[0023] in, It is a data loss function. λ is the physical residual loss function, ||λ||1 is the L1 regularization term used to improve the sparsity of the coefficients λ, and α and β are the relative weights of the physical loss function and the regularization term. Optimal network training parameters can be obtained through pre-training. and equation coefficients The specific definition of the loss function is:
[0024]
[0025] Where N d N is the number of data points in the measurement dataset used for network training. pIt is the number of residual points in the residual point dataset obtained through sampling methods such as Latin hypercube sampling. It is an approximation of the solution u of the dynamic system by a deep neural network.
[0026] Furthermore, the sparse regression process includes:
[0027] Step 4.3.1, in the residual point dataset and Upper assembly system state vector and
[0028]
[0029] in and For residual point datasets The training and validation sets were randomly divided (in a ratio of 80%:20%).
[0030]
[0031] Step 4.3.2, construct the candidate library matrix and
[0032]
[0033] Step 4.3.3, normalize the candidate library matrix and To improve the matrix condition number:
[0034]
[0035]
[0036] Step 4.3.4, set the l0 regularization parameter. and initialize error
[0037]
[0038] Step 4.3.5: Set the tolerance threshold δ1 and tolerance increment Δδ;
[0039] Step 4.3.6, once the system state vector is obtained and Candidate library matrix Φ tr and Φ va Regularization parameter γ, initial error The initial tolerance threshold δ1 and tolerance increment Δδ, according to the STRidge algorithm, have a maximum of N iterations. strSparsity identification yields activation terms and corresponding equation coefficients in the candidate library.
[0040]
[0041] Furthermore, the optimization objective of the post-training process is:
[0042]
[0043] During post-training, only the coefficients in the equation λ that are identified as activation terms by sparse regression are optimized; other terms are set to 0. This post-training process yields the optimal network training parameters. and equation coefficients
[0044] The present invention also provides a physics-based apparatus for discovering governing equations from scarce and noisy data, comprising:
[0045] The data acquisition module is used to acquire the field data u of the dynamic system, and to construct a measurement dataset by combining u with the corresponding time and space coordinate data. The residual point dataset was obtained by sampling using the Latin hypercube sampling method.
[0046] The model training module is used to train a deep neural network using a measurement dataset and a residual point dataset to obtain a pre-trained network model and equation coefficients.
[0047] The sparse identification module is used to perform sparse identification of the field variables of the dynamic system using the ridge regression algorithm with order thresholds, and to obtain the form and coefficients of the partial differential control equations of the dynamic system.
[0048] The model fine-tuning module is used to fine-tune the coefficients of the partial differential control equations of the dynamic system after identifying them, using a deep neural network; at the same time, the trained neural network is used to predict the state of the system at any time.
[0049] The present invention also provides an electronic device, including a memory and a processor, wherein the memory is coupled to the processor; wherein the memory is used to store program data, and the processor is used to execute the program data to implement the above-described physics-based method for discovering governing equations from scarce and noisy data.
[0050] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described physics-based method for discovering governing equations from scarce and noisy data.
[0051] The beneficial effects of this invention are as follows:
[0052] 1. This invention proposes a novel deep learning-based physical mechanism mining framework, which can discover the potential physical mechanism information of complex dynamic systems, and therefore has better recognition and modeling effects compared with traditional data-driven models.
[0053] 2. The method provided by this invention combines deep neural networks and sparse regression techniques. Compared with existing methods, it not only improves the prediction accuracy and applicability of the model, but also obtains the potential explicit dynamic control equations of the dynamic system. Based on the identified equations, further research on some physical mechanisms can be carried out.
[0054] 3. The method provided by this invention can effectively identify dynamic control equations even with small data sample sizes and data contaminated by noise. The method of this invention has strong robustness and universality, and can be used in general industrial systems to discover the explicit control equations of the system, thereby enabling its use for the control and guidance of practical industrial systems. Attached Figure Description
[0055] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the description of 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.
[0056] Figure 1 This is a schematic diagram of the method framework of the present invention;
[0057] Figure 2 The flowchart shows the dimensional verification process using the Navier-Stokes vorticity equation as an example.
[0058] Figure 3 This is a comparison diagram showing the full-field solutions generated by the actual Navier-Stokes vorticity equation and the governing equation discovered in this invention under three noise levels.
[0059] Figure 4 A comparison diagram showing the full-field solutions generated by the real Burgers equation and the governing equation discovered in this invention under three noise levels.
[0060] Figure 5 A comparison diagram showing the full-field solutions generated by the real heat conduction equation and the governing equation discovered in this invention under three noise levels.
[0061] Figure 6 This is a schematic diagram of an electronic device structure provided in an embodiment of the present invention. Detailed Implementation
[0062] The present invention will now be described in detail with reference to the accompanying drawings. Unless otherwise specified, the features of the following embodiments and implementations can be combined with each other.
[0063] Figure 1 This invention provides a physics-based method for discovering governing equations from scarce and noisy data. The invention has been validated on several typical complex dynamic systems, including the Burgers equation, the heat conduction equation, and the Navier-Stokes (NS) vorticity equation.
[0064] The method of the present invention specifically includes:
[0065] Step S1: Generate data.
[0066] Example: This embodiment is a two-dimensional flow system around a cylinder with local rotational dynamics. For an incompressible isotropic fluid subjected to conservative body forces, the well-known Navier-Stokes vorticity equation is as follows:
[0067] w t =-uw x -vw y +ν(w xx +w yy )
[0068] Where w is the spatiotemporal vorticity variable, {u, v} represents the fluid velocity, u and v are the velocities in the horizontal (x) and vertical (y) directions respectively, and ν represents the kinematic viscosity. The full-field solution of the Navier-Stokes vorticity equation was obtained using the submerged boundary projection method. The problem domain was discretized into a 499×199 spatial grid in the spatial domain Ω=[-1,8]m×[-2,2]m, and further discretized into 151 time steps δt=0.2s in the time domain t∈[0,30]s. The diameter L of the cylinder is 1m per unit length, and the free-flow velocity U... ∞ =1 m / s, kinematic viscosity ν = 0.01 m 2 s -1 (corresponding Reynolds number Re = U) ∞ L / ν=100).
[0069] Velocity u, v, and vorticity w were measured at 500 random spatial locations over 60 random time steps, forming the measurement dataset. The subsampling rate is 0.20% relative to the total dataset. The Latin hypercube sampling method is used to randomly sample 6 × 10⁶ samples in the spatiotemporal domain. 4 The residual points are used as the residual point dataset. Used to calculate physical residual loss.
[0070] Step S2, proceed as follows Figure 2 The dimensions are shown in the verification.
[0071] The initial overcomplete candidate library Φ consists of 300 terms, including physical constant terms (ν,ν). 2 ,1 / ν,1 / ν 2 ), polynomial terms (u,v,w,uv,uw,vw,u) 2 ,v 2 ,w 2 ), derivative term (w) x ,w y ,w xx ,w xy ,w yy ) and their combinations. Target variable w t The dimension is [1 / T] 2 (Corresponding to the number label 1 / 5) 2 After dimensional verification, Φ θ Includes 8 terms with dimensions consistent with the target variable: {w 2 ,uw x ,vw x ,uw y ,vw y ,νw xx ,νw xy ,νw yy The coefficient vector λ is initialized to zero.
[0072] Step S3: Construct a deep neural network for pre-training.
[0073] A fully connected neural network with 8 hidden layers, each containing 60 neurons, was constructed. The network has 3 input neurons for spatiotemporal coordinates and 3 output neurons for the velocity and vorticity of the flow field. The activation function is the hyperbolic tangent function tanh(·), and the network weights are initialized according to a Glorot normal distribution, with the bias initialized to zero. The pre-training process includes 5×10... 3 Adam optimization and up to 1×10 steps 4 The L-BFGS optimization process is as follows:
[0074]
[0075] in, It is a data loss function. λ is the physical residual loss function, ||λ||1 is the L1 regularization term used to improve the sparsity of the coefficients λ, and α and β are the relative weights of the physical loss function and the regularization term. Optimal network training parameters can be obtained through pre-training. and equation coefficients The specific definition of the loss function is:
[0076]
[0077]
[0078] Where N d N is the number of data points in the measurement dataset used for network training. p It is the number of residual points in the residual point dataset obtained through sampling methods such as Latin hypercube sampling. It is an approximation of the solution u of the dynamic system by a deep neural network.
[0079] Step S4: Use the STRidge algorithm to perform sparse identification on the pre-trained results and identify the control equations.
[0080] This step specifically includes the following sub-steps.
[0081] Step 4.1, in the residual point dataset and Upper assembly system state vector and
[0082]
[0083] in and For residual point datasets The training and validation sets were randomly divided (in a ratio of 8:2):
[0084]
[0085] Step 4.2, construct the candidate library matrix and
[0086]
[0087] Step 4.3, Normalize the candidate library matrix and To improve the matrix condition number:
[0088]
[0089] Step 4.4, set the l0 regularization parameter and initialize error
[0090]
[0091] Step 4.5: Set the tolerance threshold δ1 = 1.0 and the tolerance increment Δδ = 1.0;
[0092] Step 4.6, once the system state vector is obtained and Candidate library matrix Φ tr and Φ va Regularization parameter γ, initial error The initial tolerance threshold δ1 and tolerance increment Δδ are used to perform the maximum number of iterations N according to the STRidge algorithm. str Sparse identification with a value of 100 yields activation terms and corresponding equation coefficients in the candidate library.
[0093]
[0094] Step S5, the final post-training process, includes 5×10 3 The L-BFGS optimization process of the step is as follows:
[0095]
[0096] During post-training, only the coefficients in the equation λ that are identified as activation terms by sparse regression are optimized; other terms are set to 0. This post-training process yields the optimal network training parameters. and equation coefficients
[0097] The identification results of the Navier-Stokes vorticity equation for a two-dimensional cylindrical flow system under different noise levels are shown in Table 1. The relative error of the coefficients is defined as the average relative error of the identified non-zero coefficients relative to their true values. It can be observed that despite the influence of different noise levels on the data, the proposed method can still find the correct equation form with relatively small coefficient errors. Figure 3 This represents the true values of the full-field solution at different times, the predicted values of the proposed method, and the corresponding prediction errors. Figure 3 The responses at times t = [0.8, 8.0, 16.8, 21.0, 27.2] shown in the diagram were not used to train the network. Figure 3 It can be seen that the proposed method can predict vorticity patterns and magnitudes very well. With noise levels of 1%, 5%, and 10%, the l2 errors of the predicted full-field vorticity response over the entire time period are 0.58±0.48%, 1.34±0.51%, and 2.34±0.62%, respectively. This example demonstrates that the proposed method can discover closed-form Navier-Stokes vorticity equations from scarce and noisy data, exhibiting not only small equation coefficient errors but also good predictive performance and relatively small prediction errors.
[0098] Table 1: Identification results of the Navier-Stokes vorticity equation for a two-dimensional cylindrical flow system under different noise levels.
[0099]
[0100] In addition to this example, the present invention also tests two other examples: the Burgers equation and the heat conduction equation. Figure 4 A comparison diagram showing the full-field solutions generated by the real Burgers equation and the governing equation discovered in this invention under three noise levels. Figure 5 The image shows a comparison of the full-field solutions generated by the actual heat conduction equation and the control equation discovered in this invention under three noise levels. It can be seen that the method of this invention can discover the potential control equations and physical mechanisms of complex dynamic systems from noisy and scarce data, and can achieve high-precision prediction and equation identification. Therefore, this invention can provide effective technical support for the mechanism mining, prediction and control of complex dynamic systems.
[0101] The present invention also provides a physics-based apparatus for discovering governing equations from scarce and noisy data, comprising:
[0102] The data acquisition module is used to acquire the field data u of the dynamic system, and to construct a measurement dataset by combining u with the corresponding time and space coordinate data. The residual point dataset was obtained by sampling using the Latin hypercube sampling method.
[0103] The model training module is used to train a deep neural network using a measurement dataset and a residual point dataset to obtain a pre-trained network model and equation coefficients.
[0104] The sparse identification module is used to perform sparse identification of the field variables of the dynamic system using the ridge regression algorithm with order thresholds, and to obtain the form and coefficients of the partial differential control equations of the dynamic system.
[0105] The model fine-tuning module is used to fine-tune the coefficients of the partial differential control equations of the dynamic system after identifying them, using a deep neural network; at the same time, the trained neural network is used to predict the state of the system at any time.
[0106] It should be noted that the device embodiment shown in this embodiment matches the content of the above method embodiment, and the content of the above method embodiment can be referred to, and will not be repeated here.
[0107] Figure 6 This is a schematic diagram of an electronic device structure provided in an embodiment of the present invention. Please refer to... Figure 6 The electronic device provided in this embodiment includes a memory and a processor. The memory is used to store information including program instructions, and the processor is used to control the execution of the program instructions. When the program instructions are loaded and executed by the processor, they implement a physics-based method for discovering control equations from scarce and noisy data according to the present invention.
[0108] It should be noted that, in addition to Figure 6 In addition to the memory and processor shown, electronic devices may include other hardware depending on their actual functions, which will not be elaborated further.
[0109] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described physics-based method for discovering governing equations from scarce and noisy data.
[0110] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.
[0111] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0112] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0113] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.
[0114] The above embodiments are only used to illustrate the design concept and features of the present invention, and their purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. The protection scope of the present invention is not limited to the above embodiments. Therefore, all equivalent changes or modifications made based on the principles and design ideas disclosed in the present invention are within the protection scope of the present invention.
Claims
1. A physics-based method for discovering governing equations from scarce and noisy data, characterized in that, This method combines physical information neural networks with sparse regression to discover the governing equations of a dynamic system from scarce and noisy data, including the following steps: Step 1: Obtain the field data u of the dynamic system through numerical simulation or experimental observation, and construct a measurement dataset by combining u with the corresponding time and space coordinate data. Step 2: Obtain the residual point dataset by sampling using the Latin hypercube sampling method. Step 3: Construct and initialize a deep neural network, which consists of fully connected neural networks, to fit the solution to the problem and compute the candidate library Φ. θ Function terms in; Step 4: Train the deep neural network using the measurement dataset and the residual point dataset to obtain the pre-trained network model and equation coefficients; Step 5: Use the ridge regression algorithm with sequential thresholding to identify the sparsity of the field variables of the dynamic system and obtain the form and coefficients of the partial differential control equations of the dynamic system. Step 6: Once the partial differential control equations of the dynamic system are identified, the equation coefficients are fine-tuned using a deep neural network. At the same time, the trained deep neural network is used to predict the state of the system at any given time.
2. The method as described in claim 1, characterized in that, The candidate library Φ θ It was obtained by performing dimensional verification on the initial incomplete candidate library.
3. The method as described in claim 2, characterized in that, The initial overcomplete candidate library includes physical constant terms, polynomial terms, partial derivative terms, and combinations thereof.
4. The method as described in claim 1, characterized in that, The process of discovering the equation includes the following sub-steps: Step 4.1: Pre-train the deep neural network using the measurement dataset and residual point dataset, while optimizing the network parameters θ and equation coefficients λ. Step 4.2: The pre-trained network outputs the time derivative vector of the target variable. And the candidate library is calculated through automatic differentiation; Step 4.3: Perform sparse regression on the system. Term in the candidate library that is related to the system's governing equation is activated, thereby obtaining the form and coefficients of the equation. Step 4.4 involves post-training using a deep neural network to fine-tune the obtained equation coefficients and further optimize the network parameters to improve fitting accuracy.
5. The method as described in claim 4, characterized in that, The optimization objective of the pre-training process is: in, It is a data loss function. λ is the physical residual loss function, ||λ||1 is the L1 regularization term used to improve the sparsity of the coefficients λ, and α and β are the relative weights of the physical loss function and the regularization term; the optimal network training parameters are obtained through pre-training. and equation coefficients The specific definition of the loss function is: Where N d N is the number of data points in the measurement dataset used for network training. p This is the number of residual points in the residual point dataset obtained through the Latin hypercube sampling method. It contains N d The training dataset for the group data, The spatial coordinate data in this training dataset, For time coordinate data, u i This is the solution of the dynamic system corresponding to the spatiotemporal coordinates, where the subscript i is the index value, representing that it is the i-th data set in the training dataset; It contains N p The residual dataset of the group of residual points These are the spatial and temporal coordinates of the residual points, respectively. The subscript i is also an index value, representing that it is the i-th data group in the residual dataset. λ represents the data in the i-th row and j-th column of the candidate library. j Represents the j-th term in the coefficients of the equation. Represents the i-th row vector of Φ The inner product with λ, with repeated subscript j indicating summation over the traversed elements; This represents taking the partial derivative with respect to time t; It is an approximation of the solution u of the dynamic system by a deep neural network.
6. The method as described in claim 4, characterized in that, The sparse regression process includes: Step 4.3.1, in the residual point dataset Upper assembly system state vector in These are the residual point datasets. The training and validation sets were obtained by random partitioning: Step 4.3.2, construct the candidate library matrix Step 4.3.3, normalize the candidate library matrix To improve the matrix condition number: in, The normalized training candidate library Φ tr All elements in column j; Φ represents the normalized verification candidate library va All elements in the j-th column; the index: j indicates that each column is traversed and normalized. Step 4.3.4, set the l0 regularization parameter. and initialize error Step 4.3.5: Set the tolerance threshold δ1 and tolerance increment Δδ; Step 4.3.6, once the system state vector is obtained Candidate library matrix Φ tr and Φ va Regularization parameter γ, initial error The initial tolerance threshold δ1 and tolerance increment Δδ, according to the STRidge algorithm, have a maximum of N iterations. str Sparsity identification yields activation terms and corresponding equation coefficients in the candidate library.
7. The method as described in claim 4, characterized in that, The optimization objective of the post-training process is: During post-training, only the coefficients in the equation coefficient λ that are identified as activation terms by sparse regression are optimized, while other terms are set to 0. The optimal network training parameters are obtained through post-training. and equation coefficients 8. A physics-based apparatus for discovering governing equations from scarce and noisy data, characterized in that, include: The data acquisition module is used to acquire the field data u of the dynamic system, and to construct a measurement dataset by combining u with the corresponding time and space coordinate data. The residual point dataset was obtained by sampling using the Latin hypercube sampling method. The model training module is used to train a deep neural network using a measurement dataset and a residual point dataset to obtain a pre-trained network model and equation coefficients. The sparse identification module is used to perform sparse identification of the field variables of the dynamic system using the ridge regression algorithm with order thresholds, and to obtain the form and coefficients of the partial differential control equations of the dynamic system. The model fine-tuning module is used to fine-tune the coefficients of the partial differential control equations of the dynamic system after identifying them, using a deep neural network; at the same time, the trained neural network is used to predict the state of the system at any time.
9. An electronic device comprising a memory and a processor, characterized in that, The memory is coupled to the processor; wherein the memory is used to store program data, and the processor is used to execute the program data to implement a physics-based method for discovering governing equations from scarce and noisy data as described in any one of claims 1-7.
10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements a physics-based method for discovering governing equations from scarce and noisy data as described in any one of claims 1-7.