Nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and comparative learning

Abstract

The invention discloses a nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and comparative learning, and the method combines a multi-scale Koopman neural operator with physically guided sparse regression and comparative learning, and automatically discovers a control equation from chaotic high-dimensional observation data. Firstly, multi-scale representation is constructed through frequency domain decomposition and Hankel embedding, and linear modeling of nonlinear dynamics is achieved through a Koopman operator. Then, a system derivative is calculated in combination with automatic differential, a candidate function library is constructed, and a sparse explicit form of a control equation is obtained through sparse regression; in the process, a positive sample contrast learning strategy is introduced, and robustness and consistency of potential representation are ensured through enhancement means such as space cutting, noise injection and time disturbance. Aiming at the limitations of strong black box performance, poor interpretability, instability under high noise and the like in the existing method, the method integrates the advantages of operator theory, automatic differentiation, sparse regression and self-supervised learning, can consider precision, physical consistency and generalization, and has important theoretical value and engineering application significance for nonlinear dynamics mechanism discovery.

Description

Technical Field

[0001] This invention belongs to the field of industrial data-driven modeling and nonlinear dynamic system identification technology, specifically involving a nonlinear dynamic system mechanism modeling method based on multi-scale Koopman operators and contrastive learning. Background Technology

[0002] Extracting physically meaningful and interpretable system control equations from high-dimensional chaotic industrial data is a core challenge in data-driven modeling. Accurately revealing the underlying physical laws is crucial for understanding and predicting the dynamic behavior of complex systems. Traditional modeling methods typically rely on first-principles calculations and large amounts of data for dynamic inference. However, modern industrial systems are often characterized by high dimensionality, nonlinearity, and multivariable coupling, making the derivation of control equations extremely difficult. Furthermore, real-world data often contains noise and uncertainty, further increasing the difficulty of extracting robust and interpretable models from the data.

[0003] In recent years, the rapid development of big data analytics and machine learning technologies has provided new approaches to solving the aforementioned problems. Sparse regression-based equation discovery methods (such as SINDy) identify minimal itemsets by promoting sparsity, achieving a balance between model complexity and prediction accuracy to some extent. However, they still heavily rely on numerical differentiation, leading to significant errors when noise is high or data is scarce. On the other hand, while pure neural network methods possess good interpolation performance, they lack physical interpretability and have poor generalization ability to extrapolation and changes in initial conditions. Furthermore, most existing methods fail to effectively integrate multi-scale dynamic characteristics with physical constraints and struggle to handle noise and distribution shifts in real-world industrial environments. Therefore, there is an urgent need for an automatic method for discovering control equations that can balance prediction accuracy, physical interpretability, and robustness. Summary of the Invention

[0004] This invention provides a nonlinear dynamic system mechanism modeling method (PK-MCL) based on multi-scale Koopman operators and contrastive learning. By fusing multi-scale band decomposition, Koopman linearization, sparse physical regression, and self-supervised contrastive learning, it automatically identifies interpretable control equations from raw industrial data. The method first performs multi-band Fourier decomposition and Hankel embedding on the input field data, learning linear Koopman evolution operators at different scales to capture global low-frequency flows and local high-frequency details of the system. Based on this, a physical prior function library is constructed, and derivative terms are accurately calculated using automatic differentiation. Key terms and coefficients in the control equations are identified through sparse regression. Furthermore, a BYOL-style self-supervised contrastive learning mechanism is introduced, constructing positive sample pairs through spatiotemporal augmentation to enhance the model's robustness to noise and different initial conditions. Finally, by jointly optimizing the reconstruction loss, physical consistency loss, and contrastive loss, end-to-end training and automatic emergence of control equation structures are achieved.

[0005] The technical solution adopted by the present invention to achieve the above objectives is as follows:

[0006] A mechanism modeling method for nonlinear dynamic systems based on multi-scale Koopman and contrastive learning is proposed. This method captures global spatiotemporal dependencies through multi-scale Koopman neural operators and combines physics-guided sparse regression and positive sample contrastive learning strategies to automatically discover explicit governing equations that conform to physical laws from chaotic high-dimensional data. The method includes the following steps:

[0007] Data preprocessing: A random augmentation transformation is applied to the original spatiotemporal field data to generate two different augmented views of the same dynamic trajectory for subsequent comparative learning; multi-scale frequency band decomposition is performed on the data in each augmented view, and Fourier transform and truncation are performed on each frequency band.

[0008] A multi-channel parallel processing architecture is constructed: In the parallel branch, Hankel embedding is performed on each frequency band to construct a delay coordinate matrix and learn the linear Koopman evolution operator within that frequency band; the prediction results of each frequency band are fused through inverse Fourier transform and decoder to reconstruct the complete physical field prediction; a physical prior function library is constructed, the derivative term is calculated using automatic differentiation, and the key terms and their coefficients in the control equation are identified through sparse regression; thus, a parallel dual-branch architecture is obtained.

[0009] Improved loss and iterative training: Introducing BYOL-style self-supervised contrastive learning to enhance the model's robustness to noise and initial conditions by strengthening consistency constraints between views; jointly optimizing reconstruction loss, physical consistency loss and contrastive loss to achieve end-to-end training and automatic discovery of control equations.

[0010] The original spatiotemporal field data are obtained through actual on-site collection and measurement or calculated based on the concentration of substances participating in the reaction according to the kinetic differential equation.

[0011] The process of generating the enhanced view includes the following steps:

[0012] For input spatiotemporal field data Apply random spatial occlusion to simulate sensor blind spots;

[0013] Inject Gaussian noise The noise level is directly proportional to the signal energy.

[0014]

[0015] in, For the enhanced spatiotemporal field data, These are spatial coordinates. For time variables, It is the noise intensity coefficient, which controls the proportion of noise variance in the overall signal energy. Represents the original spatiotemporal field data The standard deviation is used to measure the magnitude of signal energy. Represents a Gaussian (normal) distribution;

[0016] Randomly discard or copy segments from the time series to simulate non-uniform sampling;

[0017] Apply random amplitude scaling to the data to generate an enhanced view. and .

[0018] The multi-scale frequency band decomposition and Fourier truncation include:

[0019] set up Different cutoff frequencies ,in For the encoded latent representation Perform a Fast Fourier Transform (FFT) and then adjust the values ​​based on the cutoff frequencies. The corresponding low-frequency modes are retained, and their mathematical expression is as follows:

[0020]

[0021]

[0022] in, For frequency domain representation, for or Encoded latent representation For the current frequency, yes Fourier transform results This represents the Fourier transform operator.

[0023] The process of performing Hankel embedding and learning the linear Koopman evolution operator within the frequency band includes the following steps:

[0024] Step 3-1, Hankel Delay Embedding Matrix Construction: For each frequency band Represent its frequency domain Delayed embedding is performed in the time dimension to capture spatiotemporal dynamics; the embedding dimension is defined. With delay step Construct the Hankel matrix as follows:

[0025]

[0026] in , This represents the total number of time steps.

[0027] Step 3-2, Koopman operator learning: For each frequency band Hankel matrix Divide it into the first Line as input The next line is the output. The linear Koopman operator within this frequency band is solved using the least squares method. The objective function and solution are as follows:

[0028]

[0029]

[0030] in This indicates the Moore-Penrose pseudo-inverse. It is the Frobenius norm.

[0031] The process of reconstructing a complete physical field prediction includes the following steps:

[0032] Step 4-1, Multi-step prediction and frequency band fusion: Utilizing the learned Koopman operator From the current moment latent state Start, proceed linearly Perform multi-step prediction to obtain the time step. Predicted value :

[0033]

[0034] Prediction results for each frequency band Perform inverse Fourier transform (IFFT) to reconstruct the spatial signal. And it is fused through weighted summation or attention mechanism:

[0035]

[0036]

[0037] in These are learnable band weights or weights generated by an attention mechanism. This is a fusion prediction value;

[0038] Step 4-2, High-Frequency Compensation and Final Decoding: Design a Small Convolutional Neural Network To compensate for the high-frequency details lost due to FFT truncation, the network takes the low-frequency prediction residual as input and outputs a high-frequency compensation correction term. :

[0039]

[0040] Final latent state prediction To integrate the predicted values ​​with the high-frequency compensation correction term sum:

[0041]

[0042] Finally, through the decoder Will Mapping back to physical space yields the final field prediction. .

[0043] The physical-guided sparse regression includes the following steps:

[0044] Step 5-1: Construct a candidate function library This includes, but is not limited to, polynomials, differential terms, and trigonometric functions:

[0045]

[0046] Among them, the candidate function library The parameters in the equations are used in the explicit governing equations of the physical laws.

[0047] Step 5-2: Calculate the predicted field using automatic differentiation. time derivative With spatial derivative ;

[0048] Step 5-3: Solve for the coefficient vector using sparse regression. :

[0049]

[0050] Step 5-4: Calculate the physical consistency loss. Incorporate into the total loss function;

[0051] This completes the construction of a two-branch network model for contrastive learning, wherein the two branches include parallel online networks. With the target network ;in , These are their respective network parameters.

[0052] The BYOL style comparison and learning includes the following steps:

[0053] Step 6-1: Combine the two enhanced views Enter the online network respectively With the target network ;

[0054] Step 6-2: Extract the latent representation and project it to the contrast space:

[0055]

[0056] in, , These are the online network and the target network, with the following parameters: and , and These are the projection heads for the online network and the target network, respectively. Step 6-3: Calculate the contrast loss:

[0057]

[0058] Step 6-4: Update the target network parameters using momentum:

[0059] .

[0060] The joint optimization includes the following steps:

[0061] Step 7-1, the total loss function is defined as:

[0062]

[0063] in, It is a comparison of losses. It is a loss of physical consistency;

[0064] Step 7-2: Use gradient descent algorithm to jointly optimize network parameters With sparsity coefficient ;

[0065] Step 7-3, fix during reasoning The learned Koopman operator is used for multi-step prediction and control equation extraction.

[0066] A nonlinear dynamic system mechanism modeling and control device based on multi-scale Koopman and contrastive learning includes reactor control equipment deployed at an industrial site and a host computer located in a field control room or remote cloud, wherein the host computer includes a control backend and a frontend interface.

[0067] The reactor control equipment includes, but is not limited to, a feed control valve for adjusting the amount of production materials added, an environmental control valve for adjusting the temperature and humidity of the reaction environment, and a motor for providing the force of reaction stirring, grinding, and other actions; the reactor control equipment is used to perform on-site control according to the control quantity output by the control equation generated by the host computer;

[0068] The control backend includes a memory and a processor. The memory stores programs for execution on the processor. When the program is loaded and executed, it collects and receives raw spatiotemporal field data, network model structure parameters of the dual-branch online network and the target network, training parameters, and candidate function library information from user input on the front-end interface. It then calls upon the spatiotemporal field data and network parameters to execute the method steps described above, including but not limited to constructing a dual-branch parallel Koopman neural operator network for multi-scale frequency domain decomposition, constructing Hankel embeddings, learning finite-dimensional Koopman evolution matrices for each frequency band, conducting contrastive learning training based on positive sample consistency, and solving the control equation coefficients based on the candidate function library and sparse regression. During the training iteration process, it calculates reconstruction loss, physical consistency loss, and contrastive consistency loss, and updates the online branch network and sparse coefficient matrix based on backpropagation, ultimately generating interpretable explicit control equations describing the nonlinear dynamics of the reactor. Finally, it outputs the generated control equations or the derived optimal control quantities to the reactor control equipment to achieve reaction process regulation.

[0069] The front-end interface is used to provide human-computer interaction functions. Its purposes include: collecting user-input raw spatiotemporal field observation data, network model parameters of the online network and the target network, initial values ​​of the coefficient matrix corresponding to the training hyperparameters and candidate function library, loss calculation formula for joint optimization, and other model settings; sending the above input information to the control backend, and receiving the training process visualization information output by the control backend, the feature consistency visualization results of the comparative learning dual-branch network, and the displayed control equations obtained by physical sparse regression; and displaying the modeling process and the final generated control equation expression in a visual manner on the interface to help users understand the system mechanism or execute subsequent control strategy configurations.

[0070] The present invention has the following beneficial effects and advantages:

[0071] A novel mechanism modeling method for nonlinear dynamic systems based on multi-scale Koopman and contrastive learning is proposed to automatically extract interpretable control equations from high-dimensional chaotic industrial data. This method integrates multi-scale frequency band decomposition and Koopman linearization to capture global and local dynamics, physics-guided sparse regression to identify key terms in the control equations, and BYOL self-supervised contrastive learning to enhance model robustness. The frequency domain representation obtained through multi-scale Fourier decomposition and Hankel embedding effectively separates dynamic modes at different scales, promoting the learning of linear evolution operators in each subspace. The physics-guided sparse regression module uses automatic differentiation to accurately calculate derivatives and automatically extracts key physical terms from the control equations through L1 constraints, ensuring the physical interpretability of the solution. The contrastive learning mechanism enhances the consistency constraints between views, making the model invariant to noise and changes in initial conditions, significantly improving generalization ability. Finally, through an end-to-end joint optimization framework, the explicit mathematical expression of the system control equations automatically emerges while improving prediction accuracy, providing an effective tool for modeling and interpreting complex industrial systems. Attached Figure Description

[0072] Figure 1 This is a flowchart of the method of the present invention.

[0073] Figure 2 This is a comparison of the system response between the predicted and exact solutions for an isothermal batch reactor system under different initial conditions.

[0074] Figure 3 This is a graph showing the variation of the average trajectory error of the isothermal batch reactor system over time and the error distribution at the final time.

[0075] Figure 4 This is a comparison chart of the prediction results of this invention with other advanced algorithms. Detailed Implementation

[0076] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough 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 modifications without departing from the spirit of the invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0077] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used in the description of the invention herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0078] This invention proposes a unified framework combining a physically guided Koopman operator and multi-scale comparative learning, based on the needs of dynamic modeling of complex industrial systems. The invention was experimentally validated on typical chemical reactor systems, non-isothermal continuous stirred tank reactors (CSTRs), and real grinding-classification industrial processes. Experimental results show that the invention can accurately recover nonlinear control equations under conditions of strong noise, varying initial conditions, and multi-timescale coupling, exhibiting good accuracy, robustness, and universality. Therefore, this invention can be extended to general industrial dynamic systems, enabling the automatic discovery of interpretable control equations from complex observation data and providing support for the prediction and control of actual industrial processes.

[0079] like Figure 1 The diagram shows a flowchart of an example method of this invention. A dynamic modeling method based on physics-guided Koopman operators and multi-scale contrastive learning first generates different spatiotemporal views through data augmentation and employs positive sample consistency constraints to improve the model's robustness to noise and perturbations. Then, the input data is decomposed in the frequency domain and Hankel embedded using the multi-scale Koopman neural operator (MS-KNO) to extract cross-scale global spatiotemporal dependency features, and the nonlinear dynamics of the system are captured through linear evolution and residual compensation. Further, a candidate function library is constructed using automatic differentiation, and the explicit control equations of the system are identified through physics-guided sparse regression. Finally, end-to-end optimization is performed under a joint training objective to achieve a balance between physical consistency and prediction accuracy. The program execution steps of this invention can be implemented in various programming environments such as MATLAB and Python.

[0080] The specific steps of this invention are as follows:

[0081] Step 1: This embodiment of the invention takes an isothermal batch reactor system as an example. This reactor system involves two continuous reactions. Its dynamics are governed by a set of ordinary differential equations.

[0082] (1)

[0083] (2)

[0084] (3)

[0085] in, Let A, B, and C represent the concentrations (mol / L) of species A, B, and C, respectively, and let the actual value of the reaction rate constant be set to... The initial total concentration constraint is... , .

[0086] Random spatial occlusion is applied to the raw data to simulate sensor blind spots;

[0087] Injecting Gaussian noise, the noise level is proportional to the signal energy:

[0088]

[0089] in, For the enhanced spatiotemporal field data, These are spatial coordinates (which can be coordinates in a one-dimensional, two-dimensional, or multi-dimensional region). For time variables, It is the noise intensity coefficient (noise level factor), which controls the proportion of noise variance in the overall signal energy; Represents the original spatiotemporal field data The standard deviation of the signal is used to measure the magnitude of the signal energy. It represents a Gaussian (normal) distribution.

[0090] Randomly discard or copy segments from the time series to simulate non-uniform sampling;

[0091] Apply random amplitude scaling to the data to generate an enhanced view. and .

[0092] Step 2: Perform multi-scale frequency band decomposition on the data in each enhanced view, perform Fourier transform and truncation on each frequency band, and construct a multi-channel parallel processing architecture.

[0093] Frequency band division and Fourier truncation: settings Different cutoff frequencies ,in The encoded latent representation Perform a Fast Fourier Transform (FFT) and then adjust the values ​​based on the cutoff frequencies. The corresponding low-frequency modes are retained, and their mathematical expression is as follows:

[0094]

[0095] (6)

[0096] in, For frequency domain representation, for or Encoded latent representation For the current frequency, yes Fourier transform results This represents the Fourier transform operator.

[0097] Step 3: Perform Hankel embedding for each frequency band, construct the delay coordinate matrix, and learn the linear Koopman evolution operator within that frequency band;

[0098] Step 3-1, Hankel Delay Embedding Matrix Construction: For each frequency band Represent its frequency domain Delayed embedding is performed in the time dimension to capture spatiotemporal dynamics. The embedding dimension is defined. With delay step Construct the Hankel matrix as follows:

[0099]

[0100] in , This represents the total number of time steps.

[0101] Step 3-2, Koopman operator learning: For each frequency band Hankel matrix Divide it into the first Line as input ,back Line as output The linear Koopman operator within this frequency band is solved using the least squares method. The objective function and solution are as follows:

[0102]

[0103] (9)

[0104] in This indicates the Moore-Penrose pseudo-inverse. It is the Frobenius norm.

[0105] For an input sequence of length 60, set the window length. Specifically, the delayed concentration sequences of each species are stacked to form a shape of... The Block-Hankel matrix is ​​then used. Subsequently, a single-layer nonlinear encoder is used to project this Hankel matrix into a 24-dimensional latent space.

[0106] Step 4: The prediction results of each frequency band are fused by inverse Fourier transform and decoder to reconstruct the complete physical field prediction;

[0107] Step 4-1, Multi-step prediction and frequency band fusion: Utilizing the learned Koopman operator Perform multi-step predictions. Starting from the current potential state... Start, proceed linearly Step, get the moment Predicted values:

[0108]

[0109] Prediction results for each frequency band Perform inverse Fourier transform (IFFT) to reconstruct the spatial signal. And it is fused through weighted summation or attention mechanism:

[0110]

[0111]

[0112] in These are learnable band weights or weights generated by an attention mechanism. These are the fused prediction values.

[0113] Step 4-2, High-Frequency Compensation and Final Decoding: Design a Small Convolutional Neural Network This compensates for the high-frequency details lost due to FFT truncation. The network takes the low-frequency prediction residual as input and outputs a high-frequency correction term:

[0114] (13)

[0115] The final latent state prediction is the sum of the fusion result and the high-frequency compensation:

[0116]

[0117] Finally, through the decoder Will Mapping back to physical space yields the final field prediction. .

[0118] Based on the latent space, a Koopman operator is learned offline using closed least squares regression. This allows the latent state to evolve approximately linearly over a time step. To capture fast nonlinear transient processes (especially the rapid transition from B to C in the second reaction), an additional single-layer high-frequency compensation network is introduced. This network takes the residual between the actual latent state and the Koopman linear prediction as input and outputs the corrected high-frequency transient prediction. The decoder is also a single-layer nonlinear network responsible for mapping the corrected latent state back to the physical concentration space. .

[0119] Step 5: Construct a physical prior function library, use automatic differentiation to calculate derivative terms, and identify key terms and their coefficients in the control equations through sparse regression;

[0120] Step 5-1: Construct a candidate function library It includes polynomials, differential terms, trigonometric functions, etc.

[0121] (15)

[0122] Among them, the candidate function library The parameters in the equations are used in the explicit governing equations of the physical laws.

[0123] Step 5-2: Calculate the predicted field using automatic differentiation. time derivative With spatial derivative ;

[0124] Step 5-3: Solve for the coefficient vector using sparse regression. :

[0125]

[0126] Step 5-4: Calculate the physical consistency loss. Incorporate into the total loss function. Learn a global sparse coefficient vector for all test trajectories. and apply Regularization penalty ( This promotes sparsity, ensuring that only the necessary reaction terms are identified.

[0127] This completes the construction of a two-branch network model for contrastive learning, wherein the two branches include parallel online networks. With the target network ;in , These are their respective network parameters.

[0128] Step 6: Introduce BYOL-style self-supervised contrastive learning to enhance the model's robustness to noise and initial conditions by strengthening consistency constraints between views;

[0129] Step 6-1: Randomly select two enhanced views. Enter the online network respectively With the target network ;

[0130] Step 6-2: Extract the latent representation and project it to the contrast space:

[0131]

[0132] in, , These are the online network and the target network, with the following parameters: and , and These are the projection heads for the online network and the target network, respectively.

[0133] Step 6-3: Calculate the contrast loss:

[0134] (18)

[0135] Step 6-4: Update the target network parameters using momentum:

[0136]

[0137] To enhance the model's robustness to noise and missing data, a BYOL-inspired temporal contrastive learning strategy is employed. For each training trajectory segment, 8% of time points (all channels) are randomly masked, and Gaussian noise is injected. (mol / L), and randomly discarding 5% of the frames to generate two enhanced views. Online network and target network Both methods map the enhanced Hankel sequence to a 32-dimensional projection space using a two-layer MLP. The target network parameters are updated via momentum (…). Update it.

[0138] Step 7: Jointly optimize the reconstruction loss Physical consistency loss By contrastive loss, end-to-end training and automatic discovery of control equations are achieved.

[0139] Step 7-1, the total loss function is defined as:

[0140]

[0141] in, It is a comparison of losses. It is a loss of physical consistency;

[0142] Step 7-2: Use gradient descent algorithm to jointly optimize network parameters With sparsity coefficient ;

[0143] Step 7-3, fix during reasoning The learned Koopman operator is used for multi-step prediction and control equation extraction.

[0144] In this embodiment, the setting is Using the Adam optimizer (initial learning rate) (The decay is halved every 50 rounds) Train for 200 rounds.

[0145] Therefore, the invention can effectively identify the kinetic mechanism of isothermal batch reactor systems and obtain prediction results that are highly consistent with those of the real system. Figure 2 As shown, the actual concentration change curves of a multi-reaction isothermal batch reactor system under different initial conditions are compared with the prediction results of the method proposed in this invention. The results show that the method of this invention can accurately capture the dynamic evolution trend of reactant and product concentrations. Figure 3 The invention further demonstrates the variation of prediction error over time under different initial conditions and the final steady-state error distribution. The method of the present invention maintains stable accuracy in long-term prediction, and the error is significantly lower than that of traditional methods. Figure 4 The chart comparing the prediction results of this invention with other advanced algorithms shows that this invention exhibits superior modeling accuracy and robustness. It is evident that this invention can not only accurately model idealized simulation systems but also be extended to complex industrial processes, demonstrating broad applicability and engineering application value.

[0146] In summary, this invention effectively integrates multi-scale decomposition, physical constraints, and self-supervised learning: by using multi-scale Koopman neural operators to perform frequency band decomposition and linearization modeling of the input field, it can capture global spatiotemporal dependencies and preserve dynamic features at different scales; automatic differentiation provides accurate derivative calculations, and combined with a predefined candidate function library, it provides a reliable foundation for sparse regression to identify physical mechanisms; the governing equations obtained from sparse regression not only ensure the interpretability of the equation structure but also inversely constrain network training, improving modeling accuracy and stability; simultaneously, the positive sample contrastive learning strategy ensures model consistency and robustness under different initial conditions and perturbations. By jointly optimizing the objective, the reconstruction loss, physical consistency loss, and contrastive loss are organically combined, achieving synergistic optimization of dynamic equation identification and neural network parameter learning. This framework integrates the advantages of spectral decomposition, Koopman theory, automatic differentiation, and sparse learning, demonstrating excellent effectiveness and robustness.

[0147] This invention also provides a nonlinear dynamic system mechanism modeling and control device based on multi-scale Koopman and contrastive learning, including reactor control equipment deployed at an industrial site and a host computer located in a field control room or remote cloud platform. The host computer includes a control backend and a frontend interface. The reactor control equipment includes a feed control valve for adjusting the amount of production materials added, an environmental control valve for adjusting the temperature and humidity of the reaction environment, and a motor for providing the force for reaction stirring, grinding, and other actions. The reactor control equipment is used to perform on-site control according to the control quantities output by the control equations generated by the host computer.

[0148] The control backend includes a memory and a processor. The memory stores programs that can run on the processor. When loaded and executed, the program collects and receives raw spatiotemporal field data, network model structure parameters of the dual-branch online network and the target network, training parameters, and candidate function library information from the user input on the front-end interface. It then calls upon the spatiotemporal field data and network parameters to execute the method steps described above, including but not limited to constructing a dual-branch parallel Koopman neural operator network for multi-scale frequency domain decomposition, constructing Hankel embeddings, learning finite-dimensional Koopman evolution matrices for each frequency band, conducting contrastive learning training based on positive sample consistency, and solving the control equation coefficients based on the candidate function library and sparse regression. During the training iteration process, it calculates reconstruction loss, physical consistency loss, and contrastive consistency loss, and updates the online branch network and sparse coefficient matrix based on backpropagation, ultimately generating interpretable explicit control equations describing the nonlinear dynamics of the reactor. Finally, it outputs the generated control equations or the derived optimal control quantities to the reactor control equipment to achieve reaction process regulation.

[0149] The front-end interface is used to provide human-computer interaction functions. Its purposes include: collecting user-input raw spatiotemporal field observation data, network model parameters of the online network and the target network, initial values ​​of the coefficient matrix corresponding to the training hyperparameters and candidate function library, loss calculation formula for joint optimization, and other model settings; sending the above input information to the control backend, and receiving the training process visualization information output by the control backend, the feature consistency visualization results of the comparative learning dual-branch network, and the displayed control equations obtained by physical sparse regression; and displaying the modeling process and the final generated control equation expression in a visual manner on the interface to help users understand the system mechanism or execute subsequent control strategy configurations.

[0150] The above description is merely a preferred embodiment of the present invention and does not constitute any limitation on the present invention. Any simple modifications, alterations, or equivalent structural changes made to the above embodiments based on the technical essence of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A mechanism modeling method for nonlinear dynamic systems based on multi-scale Koopman and contrastive learning (PK-MCL), characterized in that, This method captures global spatiotemporal dependencies using a multi-scale Koopman neural operator and combines physics-guided sparse regression with a positive sample contrast learning strategy to automatically discover explicit governing equations that conform to physical laws from chaotic high-dimensional data. The method includes the following steps: Data preprocessing: Apply random augmentation transformation to the original spatiotemporal field data to generate two different augmented views of the same dynamic trajectory for subsequent comparative learning; Multi-scale frequency band decomposition is performed on the data in each enhanced view, and Fourier transform and truncation are performed on each frequency band. Construct a multi-channel parallel processing architecture: In the parallel branch, Hankel embedding is performed on each frequency band to construct a delay coordinate matrix and learn the linear Koopman evolution operator in that frequency band; the prediction results of each frequency band are fused by inverse Fourier transform and decoder to reconstruct the complete physical field prediction. A physical prior function library is constructed, automatic differentiation is used to calculate derivative terms, and sparse regression is used to identify key terms and their coefficients in the control equations, resulting in a parallel dual-branch architecture. Improved loss and iterative training: Introducing BYOL-style self-supervised contrastive learning to enhance the model's robustness to noise and initial conditions by strengthening consistency constraints between views; jointly optimizing reconstruction loss, physical consistency loss and contrastive loss to achieve end-to-end training and automatic discovery of control equations.

2. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 1, characterized in that, The original spatiotemporal field data are obtained through actual on-site collection and measurement or calculated based on the concentration of substances participating in the reaction according to the kinetic differential equation.

3. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 1, characterized in that, The process of generating the enhanced view includes the following steps: For input spatiotemporal field data Apply random spatial occlusion to simulate sensor blind spots; Inject Gaussian noise The noise level is directly proportional to the signal energy. in, For the enhanced spatiotemporal field data, These are spatial coordinates. For time variables, It is the noise intensity coefficient, which controls the proportion of noise variance in the overall signal energy. Represents the original spatiotemporal field data The standard deviation is used to measure the magnitude of signal energy. Represents a Gaussian (normal) distribution; Randomly discard or copy segments from the time series to simulate non-uniform sampling; Apply random amplitude scaling to the data to generate an enhanced view. and .

4. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 1, characterized in that, The multi-scale frequency band decomposition and Fourier truncation include: set up Different cutoff frequencies ,in For the encoded latent representation Perform a Fast Fourier Transform (FFT) and then adjust the values ​​based on the cutoff frequencies. The corresponding low-frequency modes are retained, and their mathematical expression is as follows: in, For frequency domain representation, for or Encoded latent representation For the current frequency, yes Fourier transform results This represents the Fourier transform operator.

5. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 1, characterized in that, The process of performing Hankel embedding and learning the linear Koopman evolution operator within the frequency band includes the following steps: Step 3-1, Hankel Delay Embedding Matrix Construction: For each frequency band Represent its frequency domain Delayed embedding is performed in the time dimension to capture spatiotemporal dynamics; the embedding dimension is defined. With delay step Construct the Hankel matrix as follows: in , This represents the total number of time steps. Step 3-2, Koopman operator learning: For each frequency band Hankel matrix Divide it into the first Line as input The next line is the output. The linear Koopman operator within this frequency band is solved using the least squares method. The objective function and solution are as follows: in This indicates the Moore-Penrose pseudo-inverse. It is the Frobenius norm.

6. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 1, characterized in that, The process of reconstructing a complete physical field prediction includes the following steps: Step 4-1, Multi-step prediction and frequency band fusion: Utilizing the learned Koopman operator From the current moment latent state Start, proceed linearly Perform multi-step prediction to obtain the time step. Predicted value : Prediction results for each frequency band Perform inverse Fourier transform (IFFT) to reconstruct the spatial signal. And it is fused through weighted summation or attention mechanism: in These are learnable band weights or weights generated by an attention mechanism. This is a fusion prediction value; Step 4-2, High-Frequency Compensation and Final Decoding: Design a Small Convolutional Neural Network To compensate for the high-frequency details lost due to FFT truncation, the network takes the low-frequency prediction residual as input and outputs a high-frequency compensation correction term. : Final latent state prediction To integrate the predicted values ​​with the high-frequency compensation correction term sum: Finally, through the decoder Will Mapping back to physical space yields the final field prediction. .

7. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 1, characterized in that, The physical-guided sparse regression includes the following steps: Step 5-1: Construct a candidate function library This includes, but is not limited to, polynomials, differential terms, and trigonometric functions: Among them, the candidate function library The parameters in the equations are used in the explicit governing equations of the physical laws. Step 5-2: Calculate the predicted field using automatic differentiation. time derivative With spatial derivative ; Step 5-3: Solve for the coefficient vector using sparse regression. : Step 5-4: Calculate the physical consistency loss. Incorporate into the total loss function; This completes the construction of a two-branch network model for contrastive learning, wherein the two branches include parallel online networks. With the target network ;in , These are their respective network parameters.

8. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 7, characterized in that, The BYOL style comparison and learning includes the following steps: Step 6-1: Combine the two enhanced views Enter the online network respectively With the target network ; Step 6-2: Extract the latent representation and project it to the contrast space: in, , These are the online network and the target network, with the following parameters: and , and These are the projection heads for the online network and the target network, respectively. Step 6-3: Calculate the contrast loss: Step 6-4: Update the target network parameters using momentum: 。 9. The nonlinear dynamic system mechanism modeling method based on multi-scale Koopman and contrastive learning according to claim 1, characterized in that, The joint optimization includes the following steps: Step 7-1, the total loss function is defined as: in, It is a comparison of losses. It is a loss of physical consistency; Step 7-2: Use gradient descent algorithm to jointly optimize network parameters With sparsity coefficient ; Step 7-3, fix during reasoning The learned Koopman operator is used for multi-step prediction and control equation extraction.

10. A nonlinear dynamic system mechanism modeling and control device based on multi-scale Koopman and contrastive learning, characterized in that, It includes reactor control equipment deployed at the industrial site and a host computer located in the field control room or remote cloud, the host computer including a control backend and a frontend interface; The reactor control equipment includes, but is not limited to, a feed control valve for adjusting the amount of production materials added, an environmental control valve for adjusting the temperature and humidity of the reaction environment, and a motor for providing the force of reaction stirring, grinding, and other actions; the reactor control equipment is used to perform on-site control according to the control quantity output by the control equation generated by the host computer; The control backend includes a memory and a processor. The memory stores a program for running on the processor. When the program is loaded and executed, it is used to collect and receive raw spatiotemporal field data, network model structure parameters of the dual-branch online network and the target network, training parameters, and candidate function library information from the user input on the front-end interface. The spatiotemporal field data and network parameters are invoked to execute the steps of the method described in any one of claims 1-9, including but not limited to constructing a dual-branch parallel Koopman neural operator network for multi-scale frequency domain decomposition, constructing Hankel embedding, learning the finite-dimensional Koopman evolution matrix for each frequency band, conducting contrastive learning training based on positive sample consistency, and solving the control equation coefficients based on the candidate function library and sparse regression. During the training iteration, the reconstruction loss, physical consistency loss and contrast consistency loss are calculated, and the online branch network and sparse coefficient matrix are updated based on backpropagation. Finally, interpretable explicit control equations describing the nonlinear dynamics of the reactor are generated. The generated control equations or derived optimal control quantities are then output to the reactor control equipment to regulate the reaction process. The front-end interface is used to provide human-computer interaction functions. Its uses include: collecting the user-input raw spatiotemporal field observation data, network model parameters of the online network and the target network, initial values ​​of the coefficient matrix corresponding to the training hyperparameters and the candidate function library, and model settings such as the loss calculation formula for joint optimization; sending the above input information to the control backend, and receiving the training process visualization information output by the control backend, the feature consistency visualization results of the comparative learning dual-branch network, and the display control equation obtained by physical sparse regression; The modeling process and the final generated control equation expressions are displayed visually on the interface to help users understand the system mechanism or execute subsequent control strategy configurations.

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