Depth Koopman modeling method and system fused with differential quadratic programming
By integrating the deep Koopman modeling method of differentiable quadratic programming, the problem of modeling accuracy and training convergence difficulty in nonlinear dynamic systems is solved by automatically optimizing the dimension-upgrading mapping function, thus achieving efficient global linear dynamic modeling.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-01-08
- Publication Date
- 2026-04-24
AI Technical Summary
Existing Koopman modeling methods for nonlinear dynamic systems suffer from several drawbacks, including strong subjectivity in selecting the upscaling mapping function, poor modeling accuracy and generalization performance, large number of training parameters, and high convergence difficulty.
A deep Koopman modeling method integrating differentiable quadratic programming is adopted. The up-dimensional mapping function is fitted by a neural network, and the optimal high-dimensional global linear dynamic model is solved in each iteration using a differentiable quadratic programming layer. The optimality constraint relationship between the up-dimensional mapping function and the high-dimensional global linear dynamic model is established.
It improves modeling accuracy and training stability, reduces training convergence difficulty, and ensures the model's optimality and generalization performance.
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Figure CN121919692A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of dynamic modeling of nonlinear systems, and in particular to a deep Koopman modeling method and system that integrates differentiable quadratic programming. Background Technology
[0002] Currently, control and analysis methods for linear dynamic systems are relatively mature. To effectively transfer these mature methods to nonlinear dynamic systems, a global linear dynamic model needs to be constructed. This model can accurately characterize the dynamics of the original nonlinear system in a linear manner at each operating point.
[0003] The Koopman operator theory provides a fundamental theoretical framework for constructing global linear dynamic models of nonlinear dynamic systems. According to this theory, nonlinear dynamics in the original space can be mapped to linear dynamics in a higher-dimensional space using an increasing-dimensional mapping function, thereby constructing a global linear dynamic model of the original nonlinear dynamics in the higher-dimensional space.
[0004] Guided by the aforementioned theories, a series of Koopman modeling methods have emerged. These methods construct high-dimensional global linear dynamic models of nonlinear systems in a data-driven manner. Based on the algorithms employed, existing Koopman modeling methods can be mainly divided into two categories. The first category employs non-deep learning algorithms, such as the Extended Dynamic Mode Decomposition (EMD) algorithm. The basic idea of this type of method is to obtain an ascending-dimensional mapping function through manual selection, and then construct and solve a special quadratic programming problem, namely the least squares problem, based on this function, ultimately establishing the optimal high-dimensional global linear dynamic model under the ascending-dimensional mapping function. The main advantage of this type of method is that its modeling process is based on the rigorous mathematical framework of a quadratic programming problem, ensuring the optimality of the obtained high-dimensional global linear dynamic model under the ascending-dimensional mapping function. However, its disadvantages are also quite obvious: the selection of the ascending-dimensional mapping function depends on the engineer's experience, exhibiting strong subjectivity and arbitrariness; if the selection is inappropriate, it will seriously affect the modeling accuracy and generalization performance.
[0005] The second type of method employs deep learning algorithms. The basic idea of this type of method is to construct an ascending-dimensional mapping function and a high-dimensional global linear dynamic model using neural networks and trainable matrices respectively, and then achieve joint optimization of the two through an end-to-end training process. Compared with the first type of method, the advantage of this type of method is that it eliminates the need for manual selection of the ascending-dimensional mapping function; in this type of method, the ascending-dimensional mapping function can be automatically learned and continuously optimized during training. However, this type of method still has certain limitations: First, the ascending-dimensional mapping function and the high-dimensional global linear dynamic model are two independent parts to be trained, lacking the optimality constraint relationship established through quadratic programming in the first type of method, thus failing to guarantee the optimality of the obtained high-dimensional global linear dynamic model under the learned ascending-dimensional mapping function. Second, this type of method treats both the ascending-dimensional mapping function and the high-dimensional global linear dynamic model as parts to be trained, resulting in a large number of training parameters, which increases the difficulty of convergence and is detrimental to the stable training of the model. Summary of the Invention
[0006] The purpose of this invention is to provide a deep Koopman modeling method and system that integrates differentiable quadratic programming to solve the above-mentioned problems.
[0007] To achieve the above objectives, the present invention adopts the following technical solution: In a first aspect, the present invention provides a deep Koopman modeling method that integrates differentiable quadratic programming, comprising: Collect and process historical operating data, state data, and input data of nonlinear dynamic systems, and divide the data into training and validation sets; Construct a deep Koopman modeling framework incorporating a differentiable quadratic programming layer; The modeling framework is trained using historical running datasets. The parameters to be trained are optimized through training, and finally the up-dimensional mapping function of the nonlinear dynamic system being modeled is obtained, as well as the optimal high-dimensional global linear dynamic model corresponding to the function.
[0008] Furthermore, the collection and processing of historical operating data, state data, and input data of the nonlinear dynamic system, dividing the data into training and validation sets, includes: Collect historical operating data of the nonlinear dynamic system being modeled. If the system has external inputs, collect both the system's state data and input data. If the system has no external inputs, collect only the system's state data. Arrange the collected historical data in a time series format to obtain the historical operating dataset of the nonlinear dynamic system being modeled. Divide the historical operating dataset into a training set and a validation set.
[0009] Furthermore, if the system being modeled includes external inputs, the historical runtime dataset is:
[0010] (1) in This represents the historical operational dataset of the system being modeled. These represent the system state subset and the system external input subset in the dataset, respectively. and They represent the system's first... k The system's state at each time step and its external inputs; this historical running dataset contains the system's running state at M time steps and the system's external inputs at M-1 time steps; If the system being modeled does not contain external input, the historical running dataset is: This dataset contains the system operating status at M time points. The historical operating dataset is divided into a training set and a validation set: if the system being modeled includes external inputs, the training set contains the system operating state at R time points and the system external inputs at R-1 time points, and the validation set contains the system operating state at MR time points and the system external inputs at MR-1 time points; if the system being modeled does not include external inputs, the training set contains the system operating state at R time points, and the validation set contains the system operating state at MR time points.
[0011] Furthermore, the construction of the deep Koopman modeling framework incorporating a differentiable quadratic programming layer includes: A deep Koopman modeling framework integrating a differentiable quadratic programming layer is constructed. This framework includes a dimension-upgrading mapping module, a differentiable quadratic programming layer, and a linear recursion module. The dimension-upgrading mapping module is built on a neural network and is used to fit the dimension-upgrading mapping function; this module maps the state of the nonlinear dynamic system from the original space to a higher-dimensional space. The differentiable quadratic programming layer is based on the system's upgraded state output by the upgraded mapping module. If the nonlinear system being modeled has external inputs, it also solves the quadratic programming problem based on the system's external inputs to obtain the optimal high-dimensional global linear dynamic model under the current upgraded mapping function. The linear recursion module is based on the system's upgraded state at time k output by the upgraded mapping module and the high-dimensional global linear dynamic model output by the differentiable quadratic programming layer. If the nonlinear system being modeled has external inputs, it also performs system upgraded state prediction in the time domain T based on the external inputs from time k to k+T-1.
[0012] Furthermore, the dimension-upgrading mapping module is constructed based on a multilayer perceptron and an identity mapping, and is represented as follows: (2) Where Q represents the number of layers in the multilayer perceptron, and θ represents the training parameters included in the dimension-upgrading mapping module. The system's dimensionality upgrade state is output by the dimensionality upgrade mapping module; The dimension-upgrading mapping module is simplified as follows: ; If the system being modeled includes external inputs, the forward computation process of the differentiable quadratic programming layer is as follows: (3) in, and These represent the state transition matrix and input matrix of the optimal high-dimensional global linear dynamic model obtained through a differentiable quadratic programming layer under the high-dimensional mapping function fitted by the current high-dimensional mapping module. Simplified representation: (4) in, This represents the forward computation process of the differentiable quadratic programming layer; If the system being modeled does not contain external input, the forward computation process of the differentiable quadratic programming layer is as follows: (5) Simplified representation as ; Within the deep Koopman modeling framework incorporating a differentiable quadratic programming layer, this layer can automatically generate the following gradients during gradient backpropagation: if the system being modeled includes external inputs, the generated gradients include... , , as well as If the system being modeled does not contain external input, then the generated gradient includes... and During training, these gradients are used to update and optimize the training parameters θ in the dimension-upgrading mapping module. If the system being modeled includes external inputs, the linear recursion module is: (6) in, This represents the predicted value of the system's upgraded state at time k by the linear recursive module; If the system being modeled does not contain external inputs, the linear recursion module is: (7).
[0013] Furthermore, the step of training the modeling framework using historical runtime datasets, optimizing the training parameters through training, and ultimately obtaining the upgraded mapping function of the nonlinear dynamic system being modeled, as well as the optimal high-dimensional global linear dynamic model corresponding to this function, includes: The following mean absolute error (MAE) is used to validate the prediction error of the high-dimensional global linear dynamic model in the current training epoch: (9) in, T eva This indicates the length of the prediction time domain when making predictions based on the current high-dimensional global linear dynamic model during the verification process. B eva Indicates the number of prediction groups conducted during the verification process; Training will terminate when the number of training rounds reaches the preset value or the verification error is lower than the preset value. After training, the upgraded mapping function of the modeled nonlinear dynamic system is obtained, which can be expressed as: (10) Where θ* represents the training result of the parameter θ to be trained in the dimension-upgrading mapping module; Based on the upgraded mapping function obtained from training (9), and the training set, a forward operation is performed using a differentiable quadratic programming layer to output the optimal high-dimensional global linear dynamic model of the nonlinear dynamic system modeled under the upgraded mapping function, expressed as: (11) in, and These are the state transition matrix and input matrix of the model, respectively.
[0014] Furthermore, the loss function for the training process is: (8) in, L The loss function represents the training process. L 1 indicates that the obtained high-dimensional global linear dynamic model is guaranteed. T The loss function components of step prediction accuracy, L 2 represents the loss function component that guarantees the long-term prediction accuracy of the obtained high-dimensional global linear dynamic model. Describing the Frobenius norm, r It is a very small positive number.
[0015] Secondly, the present invention provides a deep Koopman modeling system that integrates differentiable quadratic programming, comprising: The data acquisition module is used to collect and process historical operating data, state data, and input data of the nonlinear dynamic system, and divide the data into training set and validation set; The model building module is used to build a deep Koopman modeling framework that incorporates a differentiable quadratic programming layer; The output module is used to train the modeling framework using historical running datasets, optimize the parameters to be trained through training, and finally obtain the up-dimensional mapping function of the nonlinear dynamic system being modeled, as well as the optimal high-dimensional global linear dynamic model corresponding to the function.
[0016] Thirdly, the present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the deep Koopman modeling method that integrates differentiable quadratic programming.
[0017] Fourthly, the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the deep Koopman modeling method that integrates differentiable quadratic programming.
[0018] Compared with the prior art, the present invention has the following technical effects: This invention employs a neural network to fit the increasing-dimensional mapping function, automatically optimizing and updating it during training, thus avoiding the subjective and blind process of manually selecting the increasing-dimensional mapping function. This effectively improves the modeling accuracy of this invention. Secondly, compared to existing deep learning-based Koopman modeling methods, this invention only uses the increasing-dimensional mapping function as the part to be trained, with a differentiable quadratic programming layer solving for the optimal high-dimensional global linear dynamic model under the current increasing-dimensional mapping function in real time during each iteration. This establishes an optimality constraint relationship between the increasing-dimensional mapping function and the high-dimensional global linear dynamic model, ensuring the optimality of the obtained model under the learned increasing-dimensional mapping function; furthermore, it reduces the number of parameters to be trained, effectively reducing the difficulty of training convergence and improving training stability and model generalization. In summary, this invention provides an effective approach for global linear dynamic modeling of nonlinear dynamic systems, possessing significant engineering application value and theoretical significance. Attached Figure Description
[0019] Figure 1 A flowchart illustrating the steps involved in incorporating a deep Koopman modeling method that integrates differentiable quadratic programming. Figure 2 A flowchart illustrating the training process in a deep Koopman modeling method that incorporates differentiable quadratic programming. Figure 3A deep learning framework diagram for a deep Koopman modeling method that integrates differentiable quadratic programming when there are external inputs to the nonlinear dynamic system being modeled. Figure 4 A deep learning framework diagram for a deep Koopman modeling method that integrates differentiable quadratic programming when there is no external input to the nonlinear dynamic system being modeled. Detailed Implementation
[0020] The present invention will be further described below with reference to the accompanying drawings: Example 1, please refer to Figure 1 This invention provides a deep Koopman modeling method that integrates differentiable quadratic programming, comprising: Collect and process historical operating data, state data, and input data of nonlinear dynamic systems, and divide the data into training and validation sets; Construct a deep Koopman modeling framework incorporating a differentiable quadratic programming layer; The modeling framework is trained using historical running datasets. The parameters to be trained are optimized through training, and finally the up-dimensional mapping function of the nonlinear dynamic system being modeled is obtained, as well as the optimal high-dimensional global linear dynamic model corresponding to the function.
[0021] The purpose of this invention is to provide a deep Koopman modeling method that integrates differentiable quadratic programming, addressing the problems of existing deep learning-based Koopman methods failing to guarantee the optimality of the obtained high-dimensional global linear dynamic model under the learned up-dimensional mapping function, and the difficulty in training convergence. This method constructs a high-dimensional global linear dynamic model of a nonlinear dynamic system in a data-driven manner.
[0022] Example 2: This invention provides a deep Koopman modeling method that integrates differentiable quadratic programming, comprising: S1. Construct the historical execution dataset of the nonlinear dynamic system being modeled, and divide it into a training set and a validation set: First, historical operational data of the nonlinear dynamic system being modeled is collected. Specifically, if the system has external inputs, both the system's state data and input data are collected simultaneously; if the system has no external inputs, only the system's state data is collected. Then, the collected historical data is arranged in a time-series format to obtain the historical operational dataset of the nonlinear dynamic system being modeled.
[0023] If the system being modeled includes external inputs, the constructed historical runtime dataset can be represented as: (1) in, This represents the historical operational dataset of the system being modeled. These represent the system state subset and the system external input subset in the dataset, respectively. and They represent the system's first... k The system's state at each time step and its external inputs. This historical runtime dataset contains the system's runtime state at M time steps and the system's external inputs at M-1 time steps.
[0024] If the system being modeled does not contain external input, then the constructed historical runtime dataset is: This dataset contains the system's operating status at M time points.
[0025] The historical operating dataset is further divided into a training set and a validation set. If the system being modeled includes external inputs, the training set contains the system operating states at R time points and the system's external inputs at R-1 time points, while the validation set contains the system operating states at MR time points and the system's external inputs at MR-1 time points. If the system being modeled does not include external inputs, the training set contains the system operating states at R time points, and the validation set contains the system operating states at MR time points.
[0026] S2. Construct a deep Koopman modeling framework incorporating a differentiable quadratic programming layer: The framework includes a dimension-upgrading mapping module, a differentiable quadratic programming layer, and a linear recursion module.
[0027] First, a dimension-upgrading mapping module is constructed. This module is used to fit the dimension-upgrading mapping function to map the state of the nonlinear dynamic system from the original space to a higher-dimensional space.
[0028] The dimension-upgrading mapping module in the above modeling framework is constructed using a multilayer perceptron and identity mapping. This module can be represented as: (2) Where Q represents the number of layers in the multilayer perceptron, and θ represents the training parameters included in the dimension-upgrading mapping module. This is the system's dimensionality upgrade state output by the dimensionality upgrade mapping module.
[0029] Furthermore, the dimension-upgrading mapping module can be simplified as follows: .
[0030] Secondly, a differentiable quadratic programming layer is constructed. This layer is based on the system's upgraded state output by the upgraded mapping module. If the nonlinear system being modeled has external inputs, the quadratic programming problem is also solved based on the external inputs of the system to obtain the optimal high-dimensional global linear dynamic model under the current upgraded mapping function.
[0031] If the system being modeled includes external inputs, the forward computation process of the differentiable quadratic programming layer can be constructed as follows: (3) in, and These represent the state transition matrix and input matrix of the optimal high-dimensional global linear dynamic model obtained through a differentiable quadratic programming layer under the high-dimensional mapping function fitted by the current high-dimensional mapping module.
[0032] Furthermore, equation (3) can be simplified to: (4) Wherein, fDQP represents the forward computation process of the differentiable quadratic programming layer.
[0033] If the system being modeled does not contain external inputs, the forward computation process of the differentiable quadratic programming layer can be constructed as follows: (5) Furthermore, equation (5) can be simplified to: .
[0034] The forward operation of the differentiable quadratic programming layer is continuously differentiable with respect to its input. Within the deep Koopman modeling framework incorporating the differentiable quadratic programming layer, this layer can automatically generate the following gradients during gradient backpropagation: if the modeled system includes external inputs, the generated gradients include... , , as well as If the system being modeled does not contain external inputs, then the generated gradients include... and During training, these gradients are used to update and optimize the parameters θ to be trained in the dimension-upgrading mapping module. These characteristics are not found in traditional quadratic programming solvers.
[0035] Finally, a linear recursive module is constructed. This module is based on the system's upgraded state at time k output by the upgraded mapping module and the high-dimensional global linear dynamic model output by the differentiable quadratic programming layer. If the nonlinear system being modeled has external inputs, it also performs system upgraded state prediction in the time domain T based on the external inputs from time k to k+T-1.
[0036] If the system being modeled includes external inputs, the linear recursive module can be constructed as follows: (6) in, This represents the predicted value of the system's upgraded state at time k by the linear recursive module.
[0037] If the system being modeled does not contain external inputs, the linear recursive module can be constructed as follows: (7) S3. A deep Koopman modeling framework incorporating a differentiable quadratic programming layer is trained based on historical running datasets to establish a high-dimensional global linear dynamic model of the nonlinear dynamic system being modeled. First, the loss function for this training is established as follows: (8) in, L The loss function represents the training process. L 1 indicates that the obtained high-dimensional global linear dynamic model is guaranteed. T The loss function components of step prediction accuracy, L 2 represents the loss function component that guarantees the long-term prediction accuracy of the obtained high-dimensional global linear dynamic model. Describing the Frobenius norm, r It is a very small positive number.
[0038] Secondly, the training process is as follows Figure 2 As shown, the details are as follows: Set the number of training rounds to 0.
[0039] 1. Based on the training set, perform forward computation using a deep Koopman modeling framework incorporating a differentiable quadratic programming layer to obtain the loss function value for the current training epoch. L ; 2. Perform gradient backpropagation on the value of the loss function to obtain the loss function. L Gradient of the training parameter θ in the dimension-upgrading mapping module ; 3. Gradient-based The Adam algorithm is used to update the training parameters θ in the dimension-upgrading mapping module. Simultaneously, the training iteration number is incremented by 1.
[0040] 4. Using the validation set, calculate the prediction error of the high-dimensional global linear dynamic model in the current training round. This error is quantified using the mean absolute error (MAE): (9) in, T eva This indicates the length of the prediction time domain when making predictions based on the current high-dimensional global linear dynamic model during the verification process. B eva This indicates the number of prediction groups conducted during the verification process.
[0041] 5. Determine whether the number of training rounds has reached the preset value or the verification error is lower than the preset value. If not, return to step 1; otherwise, end the training.
[0042] After training, the upgraded mapping function of the modeled nonlinear dynamic system can be obtained, which can be expressed as: (10) Where θ* represents the training result of the parameter θ to be trained in the dimension-upgrading mapping module.
[0043] Based on the upgraded mapping function obtained from training (9) and the training set established by S1, by performing a forward operation using a differentiable quadratic programming layer, the optimal high-dimensional global linear dynamic model of the nonlinear dynamic system modeled under this upgraded mapping function can be obtained, which can be expressed as: (11) in, and These are the state transition matrix and input matrix of the model, respectively.
[0044] This invention can automatically learn and optimize the dimensionality-upgrading mapping function, avoiding the subjective and blind process of manually selecting the dimensionality-upgrading mapping function. This effectively improves the modeling accuracy of this invention. Only the dimensionality-upgrading mapping function is used as the training component, and a differentiable quadratic programming layer solves for the optimal high-dimensional global linear dynamic model under the current dimensionality-upgrading mapping function. On the one hand, it establishes an optimality constraint relationship between the dimensionality-upgrading mapping function and the high-dimensional global linear dynamic model, thereby ensuring the optimality of the obtained model under the learned dimensionality-upgrading mapping function; on the other hand, it reduces the number of training parameters, thereby effectively reducing the difficulty of training convergence and improving training stability and model generalization.
[0045] In another embodiment of the present invention, a deep Koopman modeling system integrating differentiable quadratic programming is provided, which can be used to implement the above-mentioned deep Koopman modeling method integrating differentiable quadratic programming. Specifically, the system includes: The data acquisition module is used to collect and process historical operating data, state data, and input data of the nonlinear dynamic system, and divide the data into training set and validation set; The model building module is used to build a deep Koopman modeling framework that incorporates a differentiable quadratic programming layer; The output module is used to train the modeling framework using historical running datasets, optimize the parameters to be trained through training, and finally obtain the up-dimensional mapping function of the nonlinear dynamic system being modeled, as well as the optimal high-dimensional global linear dynamic model corresponding to the function.
[0046] The module division in this embodiment of the invention is illustrative and represents only one logical functional division. In actual implementation, other division methods may be used. Furthermore, the functional modules in the various embodiments of the invention can be integrated into a single processor, exist as separate physical entities, or be integrated into a single module. The integrated modules described above can be implemented in hardware or as software functional modules.
[0047] In another embodiment of the present invention, a computer device is provided, comprising a processor and a memory. The memory stores a computer program, which includes program instructions. The processor executes the program instructions stored in the computer storage medium. The processor may be a Central Processing Unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. It is the computing and control core of the terminal, suitable for implementing one or more instructions, specifically suitable for loading and executing one or more instructions from the computer storage medium to achieve a corresponding method flow or corresponding function. The processor described in this embodiment of the present invention can be used for the operation of a deep Koopman modeling method incorporating differentiable quadratic programming.
[0048] In another embodiment of the present invention, a storage medium is provided, specifically a computer-readable storage medium (Memory), which is a memory device in a computer device used to store programs and data. It is understood that the computer-readable storage medium here can include both the built-in storage medium in the computer device and extended storage media supported by the computer device. The computer-readable storage medium provides storage space that stores the terminal's operating system. Furthermore, the storage space also stores one or more instructions suitable for loading and execution by a processor. These instructions can be one or more computer programs (including program code). It should be noted that the computer-readable storage medium here can be high-speed RAM or non-volatile memory, such as at least one disk storage device. The processor can load and execute one or more instructions stored in the computer-readable storage medium to implement the corresponding steps of the deep Koopman modeling method integrating differentiable quadratic programming described in the above embodiments.
[0049] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention 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.
[0050] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. 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 illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0051] 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 1One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0052] 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.
[0053] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.
Claims
1. A deep Koopman modeling method integrating differentiable quadratic programming, characterized in that, include: Collect and process historical operating data, state data, and input data of nonlinear dynamic systems, and divide the data into training and validation sets; Construct a deep Koopman modeling framework incorporating a differentiable quadratic programming layer; The modeling framework is trained using historical running datasets. The parameters to be trained are optimized through training, and finally the up-dimensional mapping function of the nonlinear dynamic system being modeled is obtained, as well as the optimal high-dimensional global linear dynamic model corresponding to the function.
2. The deep Koopman modeling method integrating differentiable quadratic programming as described in claim 1, characterized in that, The process of collecting and processing historical operating data, state data, and input data of the nonlinear dynamic system, dividing the data into training and validation sets, includes: Collect historical operating data of the nonlinear dynamic system being modeled. If the system has external inputs, collect both the system's state data and input data. If the system has no external inputs, collect only the system's state data. Arrange the collected historical data in a time series format to obtain the historical operating dataset of the nonlinear dynamic system being modeled. Divide the historical operating dataset into a training set and a validation set.
3. The deep Koopman modeling method integrating differentiable quadratic programming as described in claim 2, characterized in that, If the system being modeled includes external inputs, the historical runtime dataset is: (1) in This represents the historical operational dataset of the system being modeled. These represent the system state subset and the system external input subset in the dataset, respectively. and They represent the system's first... k The state variables at any given time and the external inputs; This historical runtime dataset contains the system runtime status at M time points and the system external inputs at M-1 time points; If the system being modeled does not contain external input, the historical running dataset is: This dataset contains the system operating status at M time points. The historical operating dataset is divided into a training set and a validation set: if the system being modeled includes external inputs, the training set contains the system operating state at R time points and the system external inputs at R-1 time points, and the validation set contains the system operating state at MR time points and the system external inputs at MR-1 time points; if the system being modeled does not include external inputs, the training set contains the system operating state at R time points, and the validation set contains the system operating state at MR time points.
4. The deep Koopman modeling method integrating differentiable quadratic programming as described in claim 1, characterized in that, The construction of the deep Koopman modeling framework incorporating a differentiable quadratic programming layer includes: A deep Koopman modeling framework integrating a differentiable quadratic programming layer is constructed. This framework includes a dimension-upgrading mapping module, a differentiable quadratic programming layer, and a linear recursion module. The dimension-upgrading mapping module is built on a neural network and is used to fit the dimension-upgrading mapping function; this module maps the state of the nonlinear dynamic system from the original space to a higher-dimensional space. The differentiable quadratic programming layer is based on the system's upgraded state output by the upgraded mapping module. If the nonlinear system being modeled has external inputs, it also solves the quadratic programming problem based on the system's external inputs to obtain the optimal high-dimensional global linear dynamic model under the current upgraded mapping function. The linear recursion module is based on the system's upgraded state at time k output by the upgraded mapping module and the high-dimensional global linear dynamic model output by the differentiable quadratic programming layer. If the nonlinear system being modeled has external inputs, it also performs system upgraded state prediction in the time domain T based on the external inputs from time k to k+T-1.
5. A deep Koopman modeling method integrating differentiable quadratic programming as described in claim 4, characterized in that, The dimension-upgrading mapping module is constructed based on a multilayer perceptron and an identity mapping, and is represented as follows: (2) Where Q represents the number of layers in the multilayer perceptron, and θ represents the training parameters included in the dimension-upgrading mapping module. The system's dimensionality upgrade state is output by the dimensionality upgrade mapping module; The dimension-upgrading mapping module is simplified as follows: ; If the system being modeled includes external inputs, the forward computation process of the differentiable quadratic programming layer is as follows: (3) in, and These represent the state transition matrix and input matrix of the optimal high-dimensional global linear dynamic model obtained through a differentiable quadratic programming layer under the high-dimensional mapping function fitted by the current high-dimensional mapping module. Simplified representation: (4) in, This represents the forward computation process of the differentiable quadratic programming layer; If the system being modeled does not contain external input, the forward computation process of the differentiable quadratic programming layer is as follows: (5) Simplified representation as ; Within the deep Koopman modeling framework incorporating a differentiable quadratic programming layer, this layer can automatically generate the following gradients during gradient backpropagation: if the system being modeled includes external inputs, the generated gradients include... , , as well as If the system being modeled does not contain external inputs, then the generated gradients include... and During training, these gradients are used to update and optimize the training parameters θ in the dimension-upgrading mapping module. If the system being modeled includes external inputs, the linear recursion module is: (6) in, This indicates that the linear recursive module is for the first... k The predicted value of the system's upgraded state at any given moment; If the system being modeled does not contain external inputs, the linear recursion module is: (7)。 6. The deep Koopman modeling method integrating differentiable quadratic programming as described in claim 1, characterized in that, The process of training the modeling framework using historical runtime datasets, optimizing the parameters to be trained through training, and ultimately obtaining the upgraded mapping function of the nonlinear dynamic system being modeled, as well as the corresponding optimal high-dimensional global linear dynamic model, includes: The following mean absolute error (MAE) is used to validate the prediction error of the high-dimensional global linear dynamic model in the current training epoch: (9) in, T eva This indicates the length of the prediction time domain when making predictions based on the current high-dimensional global linear dynamic model during the verification process. B eva Indicates the number of prediction groups conducted during the verification process; Training will terminate when the number of training rounds reaches the preset value or the verification error is lower than the preset value. After training, the upgraded mapping function of the modeled nonlinear dynamic system is obtained, which can be expressed as: (10) Where θ* represents the training result of the parameter θ to be trained in the dimension-upgrading mapping module; Based on the upgraded mapping function obtained from training (9), and the training set, a forward operation is performed using a differentiable quadratic programming layer to output the optimal high-dimensional global linear dynamic model of the nonlinear dynamic system modeled under the upgraded mapping function, expressed as: (11) in, and These are the state transition matrix and input matrix of the model, respectively.
7. A deep Koopman modeling method integrating differentiable quadratic programming as described in claim 6, characterized in that, The loss function for the training process is: (8) in, L The loss function represents the training process. L 1 indicates that the obtained high-dimensional global linear dynamic model is guaranteed. T The loss function components of step prediction accuracy, L 2 represents the loss function component that guarantees the long-term prediction accuracy of the obtained high-dimensional global linear dynamic model. Denotes the Frobenius norm. r It is a very small positive number.
8. A deep Koopman modeling system integrating differentiable quadratic programming, characterized in that, include: The data acquisition module is used to collect and process historical operating data, state data, and input data of the nonlinear dynamic system, and divide the data into training set and validation set; The model building module is used to build a deep Koopman modeling framework that incorporates a differentiable quadratic programming layer; The output module is used to train the modeling framework using historical running datasets, optimize the parameters to be trained through training, and finally obtain the up-dimensional mapping function of the nonlinear dynamic system being modeled, as well as the optimal high-dimensional global linear dynamic model corresponding to the function.
9. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of a deep Koopman modeling method that integrates differentiable quadratic programming as described in any one of claims 1 to 7.
10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of a deep Koopman modeling method that integrates differentiable quadratic programming as described in any one of claims 1 to 7.