Parameterized nonlinear system identification method based on dynamic embedded GAN

Abstract

The invention relates to the technical field of deep learning, in particular to a parameterized nonlinear system identification method based on a dynamic embedded GAN. The method comprises the following steps: forming a dynamic condition generative adversarial network by adopting a deep learning method and combining a dynamical generator and a discriminator, fusing system physical parameters and response data, realizing extrapolation of nonlinear system dynamic response of a parameterized system under variable parameters, and remarkably enhancing generalization ability; meanwhile, according to the method, under the condition that initial conditions and system parameters are given, long-time-period prediction of parameterized nonlinear system dynamic response is achieved, the dependency relation of response data on system physical parameters is recognized from the response data, and recognition and prediction of any nonlinear dynamic system are achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of deep learning, and specifically relates to a method for identifying a parametric nonlinear system based on a kinetic embedding GAN. Background Art

[0002] Parameters such as the geometry, boundary conditions, and material properties of a system often have an important impact on the performance of a nonlinear dynamic system. Therefore, identifying a parametric nonlinear dynamic system is a key path for system design, control, and performance optimization. However, in the face of the problem of identifying a parametric nonlinear dynamic system, numerical calculation modeling and deep learning surrogate models are two main technical means. In practical engineering, since physical parameters characterizing the system characteristics (such as system damping) are often difficult to obtain, and a large number of numerical simulations of candidate parameter values are required to establish a complex parametric model, the numerical calculation modeling method is difficult to widely promote in practical applications. When physical knowledge is seriously insufficient and only monitoring data of dynamic responses is available, the deep learning surrogate model is an effective alternative method for modeling a parametric nonlinear dynamic system, including specific means such as control equation identification, linear embedding of nonlinear dynamics, nonlinear mode identification, and reduced-order models. These methods are mainly used for nonlinear systems with fixed parameters and cannot identify their dependence on system physical parameters from measurement data. Therefore, existing methods are difficult to solve the problem of identifying a parametric nonlinear system. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for identifying a parametric nonlinear system based on a kinetic embedding GAN: to solve the technical problem of identifying its dependence on system physical parameters from measurement data by existing methods and solving the problem of identifying a parametric nonlinear system.

[0004] The purpose of the present invention can be achieved by the following technical solutions: A method for identifying a parametric nonlinear system based on a kinetic embedding GAN, the method comprising the following steps: S1: Obtain training data for a deep learning model, where the training data includes response data of a nonlinear dynamic system and physical parameters of the nonlinear dynamic system, and the response data of the nonlinear dynamic system includes displacement data and velocity data; S2: Construct a deep learning model, where the deep learning model consists of a kinetic generator and a kinetic discriminator. The kinetic generator includes an encoder, a decoder, and a dynamic block, and substitute the training data into the deep learning model for training: S21: In the encoder , use two sets of fully connected neural networks, and use the physical parameters and the displacement and velocity of the initial state Dimensionality reduction is performed on two high-dimensional data, and the two low-dimensional data are spliced. Then, a fully connected neural network is introduced again and decomposed into the modal space displacement containing physical attribute information response and modal space velocity response, and the modal forward transformation is realized. The expression form of the encoder is as follows:

[0005] S22: In the dynamic block According to the dynamic response of the modal space at the moment , a multi-layer neural network is used for time evolution advancement to predict the response of the modal space at future moments , and the expression form of the dynamic block is as follows: ; S23: In the decoder , a single fully connected neural network is used to reconstruct the displacement and velocity response of the modal space containing system physical parameters back to the physical space, realizing the modal inverse transformation and generating the system trajectory , and the expression form of the decoder is as follows: ; S3: The system trajectory, real trajectory, and physical parameters generated by the kinetic generator are substituted into the discriminator to form a conditional generative adversarial network embedded with physical information to identify the dynamic response of the nonlinear system with variable parameters.

[0006] Furthermore, the total loss function of the kinetic generator is composed of the prediction loss and the adversarial loss ; ; Among them, represents the position corresponding to the generated trajectory point of the i-th system, represents the position corresponding to the real trajectory point of the i-th system, represents the mean square error between the position corresponding to the generated trajectory point of the i-th system and the position corresponding to the real trajectory point of the i-th system, represents the number of training samples; ; Among them, and are the weight coefficients of the prediction loss and the adversarial loss of the kinetic generator respectively.

[0007] Furthermore, the adversarial loss function of the discriminator is defined as follows: ; wherein, is the weight coefficient of the adversarial loss function of the discriminator, which is used to estimate the distribution difference between the generated system trajectory and the real trajectory.

[0008] Furthermore, in step S1, the vibration response data of the system is obtained by solving the dynamic equation as training data.

[0009] Furthermore, the non-linear dynamic system includes: a single pendulum system with one degree of freedom, a Lorenz system with multiple degrees of freedom, and a Duffing system with two degrees of freedom.

[0010] Furthermore, the identification of the dynamic response of the non-linear system with variable parameters specifically includes the identification of the dependence on different system parameters: In step S21, the variable physical parameters characterizing the system characteristics are used as the input of the encoder to obtain the dynamic response depending on the modal space of the system parameters. The dynamic response of the modal space at future moments is highly correlated with the system parameters; the decoder module is used to reconstruct back to the physical space to generate the motion trajectory depending on the system physical parameters. At the same time, a discriminator is introduced to reduce the distribution difference between the generated trajectory and the real trajectory, so as to enhance the identification of the dependence on different system parameters.

[0011] Furthermore, in step S2, a deep learning framework is constructed through the TensorFlow machine learning platform.

[0012] Compared with the existing solutions, the beneficial effects achieved by the present invention are as follows: By adopting the method of deep learning and based on the physical parameters and structural response data of the parameterized system, the present invention realizes the efficient identification of the dynamic system response of the variable system parameters. Specifically, by combining a dynamic generator and a discriminator to form a dynamic conditional generative adversarial network, integrating the system physical parameters and response data, the extrapolation of the non-linear system dynamic response of the parameterized system under variable parameters is realized, and the generalization ability is significantly enhanced; at the same time, this method realizes the long-term prediction of the dynamic response of the parameterized non-linear system under the given initial conditions and system parameters, identifies the dependence on the system physical parameters from the response data, and realizes the identification and prediction of any non-linear dynamic system.

[0013] Compared with traditional nonlinear system identification methods, this method can significantly improve the physical interpretability of the identification process and break through the limitation of only identifying nonlinear dynamic systems with fixed parameters. At the same time, this method can complete the unified prediction of the responses of a class of parameterized nonlinear dynamic systems, achieving accurate and efficient identification of nonlinear behaviors, and providing new ideas and methods for the design, control, and performance optimization of nonlinear systems in practical engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings required in the embodiments. Obviously, the drawings described below are only some embodiments recorded in the present invention. For those of ordinary skill in the art, other drawings can also be obtained based on these drawings.

[0015] Figure 1 is the flowchart of the working process of a method for identifying parameterized nonlinear systems based on kinetic embedding GAN according to an embodiment of the present invention; Figure 2 is the training flowchart of the kinetic generator according to an embodiment of the present invention; Figure 3 is the training flowchart of the discriminator according to an embodiment of the present invention; Figure 4 is the identification and prediction result diagram of the dynamic response of the pendulum system according to an embodiment of the present invention; Figure 5 is the identification and prediction result diagram of the dynamic response of the Lorenz system according to an embodiment of the present invention; Figure 6 is the identification and prediction result diagram of the response of the parameterized Duffing system with variable mass according to an embodiment of the present invention; Figure 7 is the identification and prediction result diagram of the response of the parameterized Duffing system with variable initial conditions according to an embodiment of the present invention; Figure 8 is the identification and prediction result diagram of the response of the parameterized Duffing system with variable stiffness according to an embodiment of the present invention; Figure 9 is the identification and prediction result diagram of the response of the parameterized Duffing system with variable nonlinear degree according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0016] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the protection scope of the present invention.

[0017] In addition, the described features, structures, or characteristics may be combined in any suitable manner in one or more exemplary embodiments. In the following description, numerous specific details are provided to give a thorough understanding of the exemplary embodiments of the present disclosure. However, those skilled in the art will realize that the technical solutions of the present disclosure may be practiced omitting one or more of the specific details, or other methods, components, steps, etc. may be employed. In other cases, well-known structures, methods, implementations, or operations are not shown or described in detail to avoid obscuring aspects of the present disclosure.

[0018] This embodiment provides a method for identifying a parametric nonlinear system based on a kinetic embedding GAN. Figure 1 It is a flowchart of a method for identifying a parametric nonlinear system based on a kinetic embedding GAN according to an embodiment of the present invention, as Figure 1 shown, the method includes the following steps: S1: Obtain training data for the deep learning model, where the training data includes response data of the nonlinear dynamic system and physical parameters of the nonlinear dynamic system, and the response data of the nonlinear dynamic system includes displacement data and velocity data; S2: Construct a deep learning model, where the deep learning model consists of a kinetic generator and a kinetic discriminator. The kinetic generator includes an encoder, a decoder, and a dynamic block, and substitute the training data into the deep learning model for training: S2: Construct a deep learning model, where the deep learning model consists of a kinetic generator and a kinetic discriminator. The kinetic generator includes an encoder, a decoder, and a dynamic block, and substitute the training data into the deep learning model for training: S21: In the encoder , use two sets of fully connected neural networks to reduce the dimensionality of two high-dimensional data, namely the physical parameters and the displacement and velocity of the initial state . Concatenate the two low-dimensional data, and then introduce a fully connected neural network to decompose them into the modal space displacement response containing physical attribute information and the modal space velocity response, realizing the modal forward transformation. The expression form of the encoder is as follows: ; S22: In the dynamic block , according to the dynamic response of the modal space at the moment, use a multi-layer neural network to perform time evolution and prediction of the response of the modal space at the future moment . The expression form of the dynamic block is as follows: ; S23: In the decoder , a single fully connected neural network is used to reconstruct the displacement and velocity responses in the modal space containing the system physical parameters back to the physical space, realizing modal inverse transformation and generating the system trajectory , and the expression form of the decoder is as follows: ; S3: Substitute the system trajectory, real trajectory, and physical parameters generated by the dynamics generator into the discriminator to form a conditional generative adversarial network embedded with physical information for identifying the dynamic response of a nonlinear system with variable parameters.

[0019] In summary, the present invention realizes the efficient identification of the dynamic response of a dynamic system with variable system parameters by adopting a deep learning method based on the physical parameters of a parameterized system and structural response data. Specifically, by combining a dynamics generator and a discriminator to form a dynamic conditional generative adversarial network, fusing system physical parameters and response data, the extrapolation of the nonlinear system dynamic response of a parameterized system under variable parameters is realized, and the generalization ability is significantly enhanced; at the same time, this method realizes the long-term prediction of the dynamic response of a parameterized nonlinear system under given initial conditions and system parameters, identifies the dependence of the response data on the system physical parameters from the response data, and realizes the identification and prediction of any nonlinear dynamic system.

[0020] In some embodiments, Figure 2 is the training flowchart of the dynamics generator of the embodiment of the present invention. As Figure 2 shown, based on the response data and system parameters at the current moment, the encoder is used to reduce the dimensionality of the high-dimensional data containing the system physical parameters to the modal space. In the modal space, a dynamic block is used for time evolution advancement to predict the system response in the modal space at a future moment. Finally, it is reconstructed back to the physical space through the decoder to generate the response data of the parameterized system. Among them, the total loss function of the dynamics generator consists of a prediction loss and an adversarial loss ; ; Among them, represents the position corresponding to the generated trajectory point of the i-th system, represents the position corresponding to the real trajectory point of the i-th system, represents the mean square error between the position corresponding to the generated trajectory point of the i-th system and the position corresponding to the real trajectory point of the i-th system, represents the number of training samples; ; Among them, and are the weight coefficients of the prediction loss and the adversarial loss of the kinetic generator, respectively.

[0021] In some embodiments, Figure 3 is the training flowchart of the discriminator of the embodiment of the present invention. As shown in Figure 3 , three kinds of data, namely, the system trajectory, the real trajectory s, and the system physical parameters, generated by the kinetic generator through S21 - S23 are input into the discriminator. During the feature extraction process, a conditional generative adversarial network is adopted. Each trajectory first passes through a fully connected layer, and after passing through these layers, a vector is generated; on the other hand, the physical parameters and the initial condition vector are concatenated after passing through the fully connected layer. Then, the concatenated vector is processed through several layers and finally a latent vector is generated, which is similar to the dynamic response in the modal space of the kinetic generator. Next, these two vectors (one is the vector generated by the real trajectory and the other is the vector generated by the physical and initial condition parameters) are subjected to an element-wise dot product operation. Therefore, the information of the false trajectory and the real trajectory is fused with its corresponding parameter information (this process outputs a single value). When the trajectories are flattened and passed through the fully connected layer, they will generate a single value and add it to the aforementioned single value. By applying the sigmoid function, a value between 0 and 1 is generated, which indicates whether the trajectory generated by the initial condition is a real trajectory under the condition of the given system parameters. In the above process, the system physical parameters are used as the input of the model to form a conditional generative adversarial network embedding physical information to enhance the training process of the kinetic generator. The adversarial loss function of the discriminator for identifying the dynamic response of the variable parameter nonlinear system is defined as follows: ; Among them, is the constant coefficient of the adversarial loss function of the discriminator, which is used to estimate the distribution difference between the generated system trajectory and the real trajectory.

[0022] In some embodiments, in step S1, the vibration response data of the system is obtained by solving the dynamic equation as the training data.

[0023] In some embodiments, the nonlinear dynamic system includes: a single - degree - of - freedom pendulum system, a multi - degree - of - freedom Lorenz system, and a two - degree - of - freedom Duffing system.

[0024] In some embodiments, the identification of the dynamic response of the variable parameter nonlinear system specifically includes the identification of the dependence on different system parameters: In step S21, the variable physical parameters characterizing the system characteristics are used as the encoder input to obtain the dynamic response dependent on the modal space of the system parameters. The dynamic response of the modal space at future moments is highly correlated with the system parameters. The decoder module is used to reconstruct back to the physical space to generate the motion trajectory dependent on the system physical parameters. At the same time, a discriminator is introduced to reduce the distribution difference between the generated trajectory and the real trajectory, so as to enhance the recognition of the dependence on different system parameters.

[0025] In some embodiments, in step S2, a deep learning framework is constructed through the TensorFlow machine learning platform.

[0026] In some embodiments, Figure 4 It is a graph of the recognition and prediction results of the system dynamic response according to the embodiment of the present invention; Figure 5 It is a graph of the recognition and prediction results of the system dynamic response according to the embodiment of the present invention; Figure 6 It is a graph of the recognition and prediction results of the response of the parameterized dynamic system with variable mass according to the embodiment of the present invention, Figure 7 It is a graph of the recognition and prediction results of the response of the parameterized dynamic system with variable initial conditions according to the embodiment of the present invention; Figure 8 It is a graph of the recognition and prediction results of the response of the parameterized dynamic system with variable stiffness according to the embodiment of the present invention; Figure 9 It is a graph of the recognition and prediction results of the response of the parameterized dynamic system with variable nonlinear degree according to the embodiment of the present invention. This method can better identify the nonlinear system with variable system physical parameters and has obvious advantages in terms of physical interpretability and generalization ability.

[0027] The above embodiments can be implemented in whole or in part by software, hardware, firmware, or any combination thereof. When implemented using software, the above embodiments can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions or computer programs. When the computer instructions or computer programs are loaded or executed on a computer, the processes or functions described in the embodiments of the present application are generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices. The computer instructions can be stored in a computer-readable storage medium, or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (such as infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that can be accessed by a computer or a data storage device such as a server or a data center that contains one or more collections of available media. The available medium can be a magnetic medium (such as a floppy disk, a hard disk, or a magnetic tape), an optical medium (such as a DVD), or a semiconductor medium. The semiconductor medium can be a solid-state drive.

[0028] Those of ordinary skill in the art will realize that the units and algorithm steps of the examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or in a combination of computer software and electronic hardware. Whether these functions are executed in hardware or software depends on the specific application and design constraints of the technical solution. A professional technician can use different methods to implement the described functions for each specific application, but such implementation should not be considered to exceed the scope of the present application.

[0029] Those skilled in the art can clearly understand that for the convenience and brevity of description, the specific working processes of the systems, devices, and units described above can refer to the corresponding processes in the foregoing method embodiments and will not be repeated here.

[0030] In the several embodiments provided in the present application, it should be understood that the disclosed systems, devices, and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For example, the division of the units is only for some logical function divisions, and there can be other division methods in actual implementation. For example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed couplings or direct couplings or communication connections to each other can be through some interfaces. The indirect couplings or communication connections of the devices or units can be in electrical, mechanical, or other forms.

[0031] The unit described as a separation component may or may not be physically separated. The component displayed as a unit may or may not be a physical unit, that is, it may be located in one place or may be distributed over multiple network units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0032] As described above, the above are only specific embodiments of the present application, but the protection scope of the present application is not limited thereto. Any person skilled in the art within the technical scope disclosed in the present application can easily think of changes or substitutions, which should be covered within the protection scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.

Claims

1. A parameterized nonlinear system identification method based on dynamics embedded GAN, characterized in that: The method comprises the following steps: S1: Acquire training data of a deep learning model, wherein the training data includes response data of a nonlinear dynamic system and physical parameters of the nonlinear dynamic system, and the response data of the nonlinear dynamic system includes displacement data and velocity data; S2: Build a deep learning model, where the deep learning model consists of a dynamics generator and a dynamics discriminator. The dynamics generator includes an encoder, a decoder, and a dynamic block. Substitute the training data into the deep learning model for training: S21: In encoder , using two sets of fully connected neural networks to transform the physical parameters and the displacement and velocity of the initial state The two high-dimensional data are reduced in dimension, the two low-dimensional data are concatenated, and the fully connected neural network is introduced again to decompose them into modal space displacements containing physical property information. Response and modal space velocity Response, to achieve the modal positive transformation, the encoder is expressed as follows: ； S22: In dynamic blocks In, according to Dynamic response in modal space at a given moment , using a multi-layer neural network to advance time evolution and predict the response of the modal space at future moments , the expression of dynamic blocks is as follows: ； S23: In the decoder In this paper, a single fully connected neural network is used to transform the displacement of the modal space containing the physical parameters of the system into and speed The response is reconstructed back into physical space to achieve inverse modal transformation and generate system trajectory , the decoder is expressed as follows: ； S3: Substitute the system trajectory, true trajectory and physical parameters generated by the dynamics generator into the discriminator to form a conditional generative adversarial network embedded with physical information to identify the dynamic response of nonlinear systems with variable parameters.

2. According to claim 1, a parameterized nonlinear system identification method based on dynamics embedded GAN is characterized in that: Total loss function of the dynamics generator By predicting loss and combat loss It consists of two parts: ； ； in, represents the position corresponding to the generated trajectory point of the i-th system, represents the position corresponding to the true trajectory point of the i-th system, represents the mean square error between the position corresponding to the generated trajectory point of the ith system and the position corresponding to the true trajectory point of the ith system, Indicates the number of training samples; ； in, and are the weight coefficients of the prediction loss and adversarial loss of the dynamics generator, respectively.

3. The parameterized nonlinear system identification method based on dynamics embedded GAN according to claim 1 is characterized in that: The adversarial loss function of the discriminator is defined as follows: ； in, is the weight coefficient of the adversarial loss function of the discriminator, which is used to estimate the distribution difference between the generated system trajectory and the true trajectory.

4. According to claim 1, a parameterized nonlinear system identification method based on dynamics embedded GAN is characterized in that: In step S1 , the vibration response data of the system is obtained as training data by solving the dynamic equation.

5. The parametric nonlinear system identification method based on dynamics embedded GAN according to claim 1 is characterized in that: Nonlinear dynamic systems include: single-degree-of-freedom simple pendulum system, multi-degree-of-freedom Lorentz system and two-degree-of-freedom Duffing system.

6. The parameterized nonlinear system identification method based on dynamics embedded GAN according to claim 1 is characterized in that: The identification of the dynamic response of nonlinear systems with variable parameters specifically includes the identification of the dependencies of different system parameters: In step S21, the variable physical parameters characterizing the system characteristics are used as encoder input to obtain the dynamic response that depends on the modal space of the system parameters, and predict that the dynamic response of the modal space at future moments is highly correlated with the system parameters; the decoder module is used to reconstruct back to the physical space to generate a motion trajectory that depends on the physical parameters of the system, and a discriminator is introduced to reduce the distribution difference between the generated trajectory and the real trajectory, so as to enhance the recognition of the dependency on different system parameters.

7. The parametric nonlinear system identification method based on dynamics embedded GAN according to claim 1 is characterized in that: In step S2, a deep learning framework is constructed through the Tensorflow machine learning platform.

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