Power calculation method based on spectral feature coding network

Abstract

The application discloses a dynamic calculation method based on a spectral feature coding network, comprising: analyzing a type of a structure dynamic equation corresponding to a to-be-solved dynamic problem and modal parameters; determining input variables of the structure dynamic equation, building a spectral feature coding layer, and mapping the input variables to a frequency spectrum space related to a solution of the structure dynamic equation; constructing a neural network architecture; constructing a loss function based on physical constraint conditions of the structure dynamic equation; training the neural network architecture under the constraint of the loss function, and solving the structure dynamic equation to respond to dynamic calculation. The application projects the input variables to a frequency spectrum space of the structure dynamic equation, a multilayer perceptron fits base functions with clear physical meanings in the space, effectively solves a frequency spectrum deviation problem, improves solving accuracy of a high-frequency structure dynamic equation, reduces nonlinear fitting complexity of the neural network, makes the network quickly converge, reduces iteration times, improves solving efficiency, and provides an efficient solving idea for engineering high-frequency dynamic problems.

Description

Technical Field

[0001] This invention relates to the fields of science and engineering technology, and in particular to a dynamic calculation method based on spectral feature coding networks. Background Technology

[0002] In the field of civil engineering, the calculation and solution of the dynamic response of structures are crucial for structural design, structural health monitoring, and surveillance. Currently, the mainstream approach for calculating dynamic response remains the finite element method. However, this method often consumes significant computational resources, resulting in low efficiency and difficulty in meeting engineering requirements. Existing neural network-based methods, particularly Physical Information Neural Networks (PINNs), have shown great promise. PINNs can solve relevant structural dynamic equations without any labeled data, thus calculating the dynamic response and gaining popularity. However, existing PINNs suffer from spectral bias when solving high-frequency structural dynamic equations, preferentially fitting low-frequency features and exhibiting weak fitting ability for high-frequency components. This limits their application in high-frequency problems, specifically their use in calculating dynamic response. Traditional numerical methods, such as the finite difference method and the finite volume method, can solve structural dynamic equations, but they are computationally time-consuming and highly hardware-dependent when dealing with high-frequency features.

[0003] Chinese invention patent application CN119066307A discloses a method for solving singular partial differential equations based on a hard-coded physical information neural network. Although it improves the boundary accuracy by using boundary mask hard constraint boundary conditions, it does not solve the problem of spectral deviation of the neural network for high-frequency internal oscillations.

[0004] In summary, there is an urgent need for a new technical solution for dynamic computation based on spectral feature coding networks. Summary of the Invention

[0005] The main objective of this invention is to provide a dynamic calculation method based on spectral feature coding networks, which aims to solve the technical problems of low convergence and low accuracy in solving high-frequency structural dynamic equations for describing engineering components in engineering scenarios.

[0006] To achieve the above objectives, this invention provides a dynamic calculation method based on spectral feature coding networks, applied to address the convergence and accuracy issues of high-frequency structural dynamic equations for engineering components in engineering scenarios, comprising the following steps:

[0007] S1: Analyze the type of structural dynamic equations and modal parameters corresponding to the dynamic problem to be solved. The structural dynamic equations include single-degree-of-freedom dynamic equations describing dynamic changes in a single dimension and partial differential equations of structural motion describing spatiotemporal coupled dynamic changes. For the single-degree-of-freedom dynamic equations, extract their modal parameters and corresponding general solution structures. For the partial differential equations of structural motion, decompose them into a combination of modal shape functions and a set of time-domain single-degree-of-freedom dynamic equations through variable separation.

[0008] S2: Determine the input variables of the structural dynamic equation, build a spectral feature encoding layer, and map the input variables to the spectral space related to the solution of the structural dynamic equation. The basis functions of the spectral space are determined based on the modal parameters extracted in S1.

[0009] S3: Construct a neural network architecture including the spectral feature encoding layer and a multilayer perceptron, and fit the nonlinear combination relationship of the basis functions in the spectral space through the multilayer perceptron;

[0010] S4: Construct a loss function based on the physical constraints of the structural dynamic equations. The loss function includes at least initial condition constraint terms and constraint terms of the equations themselves.

[0011] S5: Train the neural network architecture under the constraints of the loss function to directly output the solution of the single-degree-of-freedom dynamic equation; for the partial differential equation of structural motion, couple the time-domain solution obtained by training with the modal function to obtain the final solution.

[0012] Preferably, the single-degree-of-freedom dynamic equation is a linear equation describing the high-frequency vibration of the engineering structure corresponding to the dynamic problem to be solved. The modal parameters of the single-degree-of-freedom dynamic equation include at least the vibration frequency and the attenuation coefficient, and the modal parameters of the single-degree-of-freedom dynamic equation include at least the repeating conjugate root.

[0013] Preferably, the repeated conjugate root is an expression corresponding to the frequency and damping ratio of the engineering structure.

[0014] Preferably, the partial differential equation of structural motion is an equation describing multi-dimensional spatiotemporal coupled vibration, which is used at least to describe cable vibration, and the mode shape function is an orthogonal function system that satisfies the boundary conditions, thereby decoupling the spatiotemporal variables by separating the variables.

[0015] Preferably, the input variables include independent variables and equation parameters to be generalized; for high-frequency vibration problems, the input variables include at least time variables and parameters corresponding to the vibration frequency and attenuation coefficient, which include at least the structural frequency and damping ratio.

[0016] Preferably, the basis functions of the spectral feature coding layer match the general solution structure of the structural dynamic equation, and the types of the basis functions directly correspond to the modal parameters extracted in S1, including:

[0017] For periodic oscillation problems, basis functions containing trigonometric functions are used;

[0018] An exponential basis function is used to address the decay problem.

[0019] Preferably, the multilayer perceptron only performs weight learning on the basis functions in the spectral space, and achieves solution fitting through linear combination and nonlinear activation, without directly processing the original input variables.

[0020] Preferably, the initial condition constraint term is the error between the network output and the initial state of the dynamic problem, which includes at least the initial displacement and the initial velocity, and the constraint term of the equation itself is the deviation after substituting the network output into the structural dynamic equation.

[0021] Preferably, the initial condition constraint term is balanced with the constraint term of the equation itself by a preset weighting coefficient.

[0022] Preferably, the dynamic problem to be solved is a preset engineering scenario, and the analysis of the engineering scenario includes solving the structural dynamic equations.

[0023] Beneficial effects: By projecting input variables onto the spectral space of structural dynamic equations and fitting basis functions with clear physical meaning within this space using a multilayer perceptron, compared to the traditional PINN directly fitting the original input, this method not only effectively solves the spectral bias problem and significantly improves the solution accuracy of high-frequency structural dynamic equations, but also greatly reduces the nonlinear fitting complexity of the neural network, enabling the network to converge quickly and reducing the number of iterations to improve solution efficiency. Furthermore, for generalizing single-degree-of-freedom dynamic equations, incorporating the parameters of the equation to be generalized into the input accurately captures the frequency multi-scale oscillation characteristics caused by parameter changes, maintaining high-precision solutions over a wide frequency range, reducing dependence on a large number of parameter samples, and enhancing the applicability and robustness of solving structural dynamic equations corresponding to engineering structures. In addition, extending the method to the solution of partial differential equations of structural motion based on variable separation effectively captures high-frequency spatiotemporal coupling characteristics, overcoming the limitations of traditional PINN in solving high-frequency partial differential equations of structural motion, and providing an efficient solution approach for high-frequency dynamic problems in engineering applications. Attached Figure Description

[0024] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0025] Figure 1 A flowchart illustrating the dynamic calculation method based on spectral feature coding network provided in an embodiment of the present invention;

[0026] Figure 2 A schematic diagram illustrating the vibration problem of a three-dimensional truss provided in an embodiment of the present invention;

[0027] Figure 3 A schematic diagram of a single-degree-of-freedom system in modal space provided in an embodiment of the present invention;

[0028] Figure 4 The dynamic response time history curves using the SFE-PINN method;

[0029] Figure 5 The dynamic response time history curves using the Vanilla PINN method;

[0030] Figure 6 The dynamic response time history curves using the PIRBN method;

[0031] Figure 7 The dynamic response time history curves using the KINN method;

[0032] Figure 8 The corresponding point-by-point absolute error curves using the SFE-PINN, PIRBN, and KINN methods are provided for embodiments of the present invention;

[0033] Figure 9 A schematic diagram of forced vibration of a string provided in an embodiment of the present invention;

[0034] Figure 10 The numerical calculation of the wave equation provided in the embodiment of the present invention is shown; where (a) is the numerical solution of the wave equation and (b) is the absolute error of the numerical solution of the wave equation.

[0035] Figure 11 The following is a description of the wave equation calculation based on SFE-PINN provided in an embodiment of the present invention; wherein, (a) is the wave equation calculation result based on SFE-PINN provided in an embodiment of the present invention, and (b) is the absolute error of the wave equation calculation result based on SFE-PINN;

[0036] Figure 12The calculation results of the wave equation based on PIRBN provided in the embodiments of the present invention are shown in (a) and (b) are the absolute errors of the wave equation calculation results based on PIRBN.

[0037] Figure 13 The calculation of the wave equation based on KINN is provided in the embodiment of the present invention; wherein, (a) is the calculation result of the wave equation based on KINN, and (b) is the absolute error of the calculation result of the wave equation based on KINN.

[0038] The implementation, functional features, and advantages of this invention will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0039] It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of the invention.

[0040] In the fields of science and engineering, the efficient solution of high-frequency structural dynamic equations used to describe engineering scenarios and their components has always been a key challenge.

[0041] Existing Physical Information Neural Networks (PINNs) suffer from spectral bias when solving high-frequency structural dynamic equations. They preferentially fit low-frequency features and have a weak ability to fit high-frequency components, which limits their application in high-frequency problems, i.e., their application in dynamic calculations of engineering scenarios and engineering components.

[0042] Traditional numerical methods, such as the finite difference method and the finite volume method, can solve structural dynamic equations, but they are computationally time-consuming and hardware-dependent when dealing with high-frequency features, which limits their application in the dynamic calculation of engineering scenarios and engineering components.

[0043] To address the aforementioned technical problems, this embodiment presents a Spectral Feature Encoding Physics Information Neural Network (SFE-PINN). By explicitly embedding the spectral information of the structural dynamic equations into the neural network architecture, the network's ability to model high-frequency and multi-scale oscillations is enhanced, solving the problems of convergence difficulty and low accuracy in solving high-frequency structural dynamic equations using existing methods.

[0044] Reference Figure 1 , Figure 1 This is a flowchart of the dynamic calculation method based on spectral feature coding network provided in this embodiment.

[0045] like Figure 1As shown, this embodiment discloses a dynamic calculation method based on spectral feature coding networks, applied to the convergence and accuracy problems of high-frequency structural dynamic equations for engineering components in engineering scenarios, including the following steps:

[0046] S1: Analyze the type of structural dynamic equations and modal parameters corresponding to the dynamic problem to be solved. The structural dynamic equations include single-degree-of-freedom dynamic equations describing dynamic changes in a single dimension and partial differential equations of structural motion describing spatiotemporal coupled dynamic changes. For the single-degree-of-freedom dynamic equations, extract their modal parameters and corresponding general solution structures. For the partial differential equations of structural motion, decompose them into a combination of modal vibration functions and a set of time-domain single-degree-of-freedom dynamic equations through variable separation.

[0047] Specifically, the single-degree-of-freedom dynamic equation is a linear equation describing the high-frequency vibration of the engineering structure corresponding to the dynamic problem to be solved. The modal parameters of the single-degree-of-freedom dynamic equation include at least the vibration frequency and the attenuation coefficient, and the modal parameters of the single-degree-of-freedom dynamic equation include at least the repeated conjugate roots.

[0048] Specifically, the repeated conjugate roots are expressions corresponding to the frequency and damping ratio of the engineering structure.

[0049] It should be noted that the characteristic roots of constant coefficient differential equations include simple real roots, complex conjugate roots, multiple real roots, and repeated conjugate roots. Specifically: simple real roots correspond to exponentially decaying structures; complex conjugate roots correspond to periodically oscillating structures; and multiple real roots and repeated conjugate roots correspond to composite structures containing polynomial factors. For the modal parameters of single-degree-of-freedom dynamic equations, which include repeated conjugate roots, this embodiment selects repeated conjugate roots to correspond to the frequency and damping ratio of the engineering structure.

[0050] In practical engineering applications, we first determine the corresponding structural dynamic equations based on the dynamic problem to be solved. Then, we use the equations corresponding to the dynamic problem to be solved... Taking the dynamic equation of a structure with constant coefficients as an example, we obtain equation (1):

[0051] Equation (1)

[0052] in, The coefficients of the equation are... and These are the dependent and independent variables of the equation, respectively. For non-homogeneous terms, It is the order of the derivative.

[0053] For equation (1), we obtain the corresponding homogeneous equation, which is expressed as equation (2):

[0054] Equation (2)

[0055] For equation (2), its characteristic equation is expressed as equation (3):

[0056] Equation (3)

[0057] Therefore, we can classify the corresponding eigenvalues ​​into the following four categories:

[0058] Type I: Simple real roots, the characteristic equation has simple real roots. Then the general solution of the homogeneous equation corresponds to equation (4):

[0059] Equation (4)

[0060] in, The constants determined by the boundary conditions.

[0061] Type II: Complex conjugate roots, where the roots of the characteristic equation are a pair of simple complex conjugate roots. Then the general solution of the homogeneous equation corresponds to equation (5):

[0062] Equation (5)

[0063] in, and Let be a constant determined by the boundary conditions. and It is a coefficient related to frequency and damping ratio.

[0064] Type III: Multiple real roots, the characteristic equation root is... , heavy roots Then the general solution of the homogeneous equation corresponds to equation (6):

[0065] Equation (6)

[0066] in, , , ..., Let be a constant determined by the boundary conditions.

[0067] Type IV: Repeated conjugate roots, the roots of the characteristic equation are Repeated conjugate roots Then the general solution of the homogeneous equation corresponds to equation (7):

[0068] Equation (7)

[0069] in, and Let be a constant determined by the boundary conditions.

[0070] Specifically, the partial differential equation of structural motion is an equation describing multi-dimensional spatiotemporal coupled vibration. This equation is used to describe at least cable vibration. The mode shape function is an orthogonal function system that satisfies the boundary conditions. Spatiotemporal variables are decoupled by separating variables.

[0071] In practical engineering applications, the equations used to describe cable vibration are partial differential equations of structural motion. Therefore, variable separation is performed on them. Taking the two-dimensional linear partial differential equation of motion (PDE) corresponding to equation (8) as an example,

[0072] Equation (8)

[0073] in, and For partial differential operators of the space and time terms, It is a non-homogeneous term. Its boundary condition is given by equation (9):

[0074] Equation (9)

[0075] in, It is the order of the time derivative.

[0076] The solution of PDEs and non-homogeneous terms All according to the characteristic function system Spatiotemporal separation yields equation (10):

[0077] Equation (10)

[0078] Substituting it into equation (8), we get equation (11):

[0079] Equation (11)

[0080] Among them, coefficient Depends on the spatial differential operator The specific form of. Among them, the first The set of differential equations is given by equation (12):

[0081] Equation (12)

[0082] The initial condition of the equation is given by equation (13):

[0083] Equation (13)

[0084] Following the steps for handling structural dynamic equations with constant coefficients, the above-mentioned structural dynamic equations are subjected to characteristic analysis and processing.

[0085] S2: Determine the input variables of the structural dynamic equation, build a spectral feature encoding layer, and map the input variables to the spectral space related to the solution of the structural dynamic equation. The basis functions of the spectral space are determined based on the modal parameters extracted in S1.

[0086] S3: Construct a neural network architecture that includes a spectral feature encoding layer and a multilayer perceptron, and fit the nonlinear combination relationship of basis functions in the spectral space through the multilayer perceptron.

[0087] S4: Construct a loss function based on the physical constraints of the structural dynamic equations. The loss function should include at least the initial condition constraint terms and the constraint terms of the equations themselves.

[0088] Specifically, the input variables include independent variables and equation parameters to be generalized; for high-frequency vibration problems, the input variables include at least time variables and parameters corresponding to the vibration frequency and attenuation coefficient, which include at least the structural frequency and damping ratio.

[0089] Specifically, the basis functions of the spectral feature coding layer are matched with the general solution structure of the structural dynamic equation, and the types of the basis functions directly correspond to the modal parameters extracted by S1, including:

[0090] For periodic oscillation problems, basis functions containing trigonometric functions are used;

[0091] An exponential basis function is used to address the decay problem.

[0092] Specifically, the multilayer perceptron only learns the weights of the basis functions in the spectral space and achieves the fitting of the solution through linear combination and nonlinear activation, without directly processing the original input variables.

[0093] Specifically, the initial condition constraint term is the error between the network output and the initial state of the dynamic problem. The initial state of the dynamic problem includes at least the initial displacement and the initial velocity. The constraint term of the equation itself is the deviation after substituting the network output into the structural dynamic equation.

[0094] Specifically, the initial condition constraints are balanced with the constraints of the equation itself through preset weighting coefficients.

[0095] In practical applications, the choice of input variables for structural dynamic equations depends on the specific problem. If only a single-degree-of-freedom dynamic equation (ODE) is solved, the input variable of the neural network is the independent variable that is used to solve the ODE. Taking the ODE shown in equation (1) above as an example, its input variable is the time variable. To solve a class of ODEs, i.e., to generalize some of the coefficients of ODEs, it is necessary to add input variables. Taking a single-degree-of-freedom dynamic equation with a double-repeating eigenvalue as an example, the coefficient generalization of the corresponding second-order non-homogeneous constant-coefficient ODE is shown in equation (14):

[0096] Equation (14)

[0097] The characteristic equation and characteristic roots corresponding to equation (14) are given by equation (15):

[0098] Equation (15)

[0099] Equation (15) is equivalent to equation (16):

[0100] Equation (16)

[0101] Based on equation (16), then... , and As input parameters of the neural network, and It is a coefficient related to frequency and damping ratio. It is a time variable.

[0102] After determining the input variables of the neural network, a spectral feature encoding layer is constructed based on the constant coefficient ODE to be solved or the feature information of the ODE after separating the PDE variables. This layer projects the input variables onto the spectrum, thereby assisting the neural network in extracting spectral information. For a non-homogeneous constant coefficient ODE, the general solution of its structural dynamic equation can be described as equation (17):

[0103] Equation (17)

[0104] in, This is a particular solution to the equation for a single-degree-of-freedom free vibration.

[0105] For a particular solution of the equation of free vibration with one degree of freedom Its specific form is the same as the non-homogeneous term of the equation. Closely related. Typically, the form of a particular solution can be assumed to be... A certain functional relationship is determined by substituting it into the original equation and solving it. Furthermore, the general solution of non-homogeneous ODEs can be expressed as equation (18):

[0106] Equation (18)

[0107] in, express and This indicates a certain functional relationship between the terms. Therefore, the general solution of the original equation is a linear superposition of a function concerning the non-homogeneous terms and the terms of the general solution of the homogeneous equation.

[0108] Starting from the structure of the ODE solution as shown in equation (17), a spectral feature coding layer is set up specifically according to the type of ODE eigenvalues. Taking a second-order non-homogeneous constant-coefficient ODE as an example:

[0109] When the eigenvalue is a single real root corresponding to class I, the spectral feature coding layer is designed according to the structure of the ODE solution as shown in equation (19):

[0110] Equation (19)

[0111] in, and These are the two real roots of the characteristic equation.

[0112] When the eigenvalues ​​are double real roots corresponding to Class III, the spectral feature coding layer is designed as equation (20):

[0113] Equation (20)

[0114] in, is a double real root of the characteristic equation.

[0115] When the eigenvalues ​​are the double-repeated roots corresponding to class IV, the spectral feature coding layer is designed as equation (21):

[0116] Equation (21)

[0117] in, and It is a coefficient related to frequency and damping ratio. It is a time variable.

[0118] Based on the above encoding, we perform spectral projection on the input variables using the feature encoding layer, expressing the input variables as spectral basis functions that are highly correlated with the solution of the structural dynamic equation. Furthermore, a multilayer perceptron is connected after the feature encoding layer to fit the nonlinear combination relationship between the basis functions.

[0119] It should be noted that, through the aforementioned spectral projection process, the basis functions incorporate the spectral information of the ODE. Since the multilayer perceptron primarily operates within the spectral space of the ODE, this network architecture can capture the spectral information of the ODE more comprehensively and accurately, exhibiting stronger robustness to high-frequency structural dynamic equations. Furthermore, because the spectral features related to the solution of the structural dynamic equations are all represented by the basis functions in the feature encoding layer, the multilayer perceptron only needs to fit the nonlinear relationships between the basis functions, greatly reducing the burden on the multilayer perceptron and thus improving the network's training efficiency.

[0120] After establishing the basic architecture of the neural network, a loss function needs to be constructed based on the physical information of the structural dynamic equations to guide the training process of the neural network. The input of the neural network is the independent variable and its related parameters (for ODE generalization), and the output is the solution of the ODE. The mathematical expressions of the loss function are equations (22) to (24):

[0121] Equation (22)

[0122] Equation (23)

[0123] Equation (24)

[0124] in, and These are the initial conditions of the ODE and the loss function constructed by the ODE itself, respectively, which represent the physical information embedded in the neural network.

[0125] S5: Train the neural network architecture under the constraint of the loss function to directly output the solution of the single-degree-of-freedom dynamic equation; for the partial differential equation of structural motion, couple the time-domain solution obtained by training with the modal function to obtain the final solution.

[0126] Specifically, when solving the partial differential equations of structural motion, the numerical solution is calculated by superimposing a finite number of truncated modal functions, and the number of truncated terms is adjusted based on the accuracy requirements.

[0127] Under the constraints of physical information, a neural network for training spectral feature encoding physical information is used to obtain the solution to the ODE. If the structural dynamic equation to be solved is an ODE, the output of the neural network is the desired result; if the equation to be solved is a PDE, the calculated ODE solution needs to be coupled with the feature function system. Specifically, the calculated solution is coupled with the feature function system... Substituting into equation (10) yields the solution to the PDE.

[0128] Based on the above, this embodiment combines the general solution structure of the constant coefficient dynamic equation with the method of separation of variables to construct a feature encoding layer, which can effectively map the input variables to the spectral space, thereby reducing the nonlinear fitting complexity of the neural network.

[0129] Specifically, the dynamic problem to be solved is a pre-defined engineering scenario, and the analysis of this scenario includes solving the structural dynamic equations. For example, in practical problems such as elastic wave propagation and cable vibration, SFE-PINN can solve the relevant structural dynamic equations more efficiently, promoting the development of numerical simulation technology in these fields.

[0130] In the specific application of this embodiment, we conducted a comparative experiment on the ability of SFE-PINN to capture high-frequency dynamic responses under external load conditions with traditional PINN (Vanilla PINN), PIRBN, and KINN based on equation (25). Figure 2 Taking the vibration problem of the three-dimensional truss shown as an example, the dynamic response of the truss under load F is calculated based on the modal superposition method, and the truss is projected from a multi-degree-of-freedom system in physical space onto a model such as... Figure 3 The diagram shows a single-degree-of-freedom system in modal space. For ease of explanation, the three-dimensional truss results are for illustrative purposes only and do not represent specific dimensions. It is assumed that the dynamic response of a single degree of freedom in modal space can be described by the following equation: ODE.

[0131] Equation (25)

[0132] Among them, damping ratio natural frequency External load The non-homogeneous terms of the equation have only a single frequency scale.

[0133] Reference Figures 4 to 8 , Figures 4 to 8 The solution results for the single-frequency forced vibration problem are shown, where, Figure 4 The dynamic response time history results curves using the SFE-PINN method are shown. Figure 5 The dynamic response time history results curves using the Vanilla PINN method are shown. Figure 6 The dynamic response time history results curves using the PIRBN method are shown. Figure 7 The dynamic response time history results curve using the KINN method, Figure 8 The corresponding point-by-point absolute error curve is given.

[0134] like Figures 4 to 8 As shown, under a single-frequency external load, the prediction curve of SFE-PINN closely matches the numerical solution, while Vanilla PINN, unable to capture high-frequency oscillation modes, exhibits a significant deviation from the numerical solution. Although PIRBN and KINN can approximate the response trend, their amplitude errors are significantly higher than those of SFE-PINN. To further evaluate the advantages of each algorithm, Figure 8 The error evolution curves of SFE-PINN, PIRBN, and KINN over time are presented. It should be noted that VanillaPINN is completely ineffective in solving this equation, so its error curve is not shown in the figure. Analysis of the error evolution curves reveals that although SFE-PINN's accuracy in solving the dynamic response under single-frequency out-of-scale load conditions is slightly lower than PIRBN near the right boundary of the time domain, overall, SFE-PINN's error is lower than the other two algorithms throughout the entire time domain. Especially in the initial stage, SFE-PINN's error is only about half that of PIRBN and about one-third that of KINN.

[0135] In the specific application of this embodiment, for high-frequency constant coefficient PDEs, we consider as follows: Figure 9The dynamic behavior of a 1m long string fixed at both ends under external load can be described by the two-dimensional wave equation as shown in equation (26):

[0136] Equation (26)

[0137] The initial boundary value conditions of the equation are given by equation (27):

[0138] Equation (27)

[0139] in, , Reference wave velocity parameters , source item The solutions are obtained based on SFE-PINN, PIRBN, and KINN, respectively, with the true solution of the PDE obtained using the finite difference method. It should be noted that PIRBN and KINN directly embed the PDE for solution, while SFE-PINN solves the corresponding single-degree-of-freedom dynamic equations using the method of separation of variables.

[0140] Reference Figures 10 to 13 ,in, Figure 10 In the middle (a), the numerical solution of the wave equation is shown. Figure 10 In the middle (b), the absolute error of the numerical solution of the wave equation is represented. Figure 11 (a) shows the calculation results of the wave equation based on SFE-PINN. Figure 11 In the middle (b), the absolute error of the wave equation calculation results based on SFE-PINN is represented. Figure 12 In the middle (a), the wave equation calculation results based on PIRBN are shown. Figure 12 In the middle (b), the absolute error of the wave equation calculation results based on PIRBN is shown. Figure 13 (a) shows the calculation results based on the wave equation using KINN. Figure 13 In the middle (b), the absolute error of the wave equation calculation results based on KINN is represented.

[0141] like Figures 10 to 13 The solution results for this PDE are shown in the figure. Figure 10 The solution surface of the PDE reveals that, due to the presence of non-homogeneous source terms, the solution surface exhibits a spatiotemporally coupled oscillation mode. The results show that PIRBN and KINN are completely ineffective in handling this complex spatiotemporally coupled oscillation mode, while the solution obtained by SFE-PINN is closest to the numerical solution, with a maximum pointwise absolute error of only [value missing]. This advantage can be attributed to the decoupling of spatiotemporal coupled vibration features by the separate variable calculation mode employed in this embodiment. Specifically, the oscillation modes in the spatial domain are characterized by characteristic functions, while the high-frequency vibrations in the time domain are calculated based on the SFE-PINN feature encoding framework.

[0142] In summary, the dynamic calculation method based on spectral feature coding networks in this embodiment proposes at least the following technical solutions to address the technical problems of difficult convergence and low accuracy in solving high-frequency structural dynamic equations in the prior art:

[0143] (1) The SFE-PINN architecture is proposed, which explicitly encodes the spectral feature information of the structural dynamic equation into the neural network, effectively alleviating the spectral bias problem of the traditional PINN when solving high-frequency structural dynamic equations and significantly improving the modeling ability of high-frequency oscillation modes. This method can make full use of the structural information of the structural dynamic equation itself, so that the neural network focuses more on fitting high-frequency features during the learning process, thus performing well in solving high-frequency problems.

[0144] (2) By combining the general solution structure of the constant coefficient dynamic equation and the method of separation of variables, a feature encoding layer is constructed to reduce the nonlinear fitting complexity of the neural network, enabling the network to converge quickly and obtain a high-precision approximate solution. By decomposing the complex high-frequency oscillation mode into a linear superposition of basis functions, the nonlinear relationship that the network needs to learn is greatly reduced, improving training efficiency and solution accuracy.

[0145] (3) When processing the generalization calculation of ODE coefficients, the ODE equation coefficients are embedded as input into the feature encoding layer to achieve adaptive response to changes in spectral features under different coefficient combinations, thus exhibiting good generalization ability and robustness. This means that when faced with changes in equation characteristics caused by coefficient changes, SFE-PINN can quickly adapt and solve the problem accurately without redesigning the network structure or performing large-scale retraining.

[0146] (4) SFE-PINN was successfully extended to the solution of partial differential equations of structural motion. Based on the method of separation of variables, it effectively captured spatiotemporal coupled oscillation modes, providing an efficient and accurate method for solving complex problems such as high-frequency wave equations. This method breaks through some limitations of traditional neural networks in dealing with spatiotemporal coupled problems and can more flexibly cope with complex physical phenomena.

[0147] Based on the above technical solution, this embodiment achieves at least the following technical effects:

[0148] (1) By projecting the input variables into the spectrum space of the structural dynamic equation and fitting the basis functions in the spectrum based on the multilayer perceptron, the solution accuracy of the high-frequency structural dynamic equation is significantly improved compared with the traditional PINN directly fitting the input variables.

[0149] (2) By directly fitting the basis functions with more physical meaning in the spectrum space, the nonlinear fitting complexity of the neural network is reduced, enabling the network to converge quickly and reach a stable state in a smaller number of iterations, thus improving the solution efficiency.

[0150] (3) In the generalization solution of single-degree-of-freedom dynamic equations, it has better generalization ability and robustness, can accurately capture the frequency multi-scale oscillation characteristics caused by coefficient changes, and maintain high-precision solutions over a wide frequency range. This is of great significance for solving structural dynamic equations with uncertain parameters in engineering practice, as it can reduce the dependence on a large number of parameter samples and improve the applicability of the model.

[0151] (4) Successfully expanded the application scope of PINN, enabling it to effectively solve PDEs with high-frequency spatiotemporal coupling characteristics, providing new solutions for high-frequency dynamic problems in practical engineering and physics. For example, in practical problems such as elastic wave propagation and cable vibration, SFE-PINN can solve relevant structural dynamic equations more efficiently, promoting the development of artificial intelligence technology in these fields.

[0152] It should be understood that the above are merely illustrative examples and do not constitute any limitation on the technical solutions of the present invention. In specific applications, those skilled in the art can make settings as needed, and the present invention does not impose any restrictions on this.

[0153] It should be noted that the workflow described above is merely illustrative and does not limit the scope of protection of this invention. In practical applications, those skilled in the art can select some or all of the workflow to achieve the purpose of this embodiment according to actual needs, and no restrictions are imposed here.

[0154] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.

[0155] Through the above description of the embodiments, those skilled in the art can clearly understand that the methods of the above embodiments can be implemented by means of software plus necessary general-purpose hardware platforms. Of course, they can also be implemented by hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium (such as read-only memory / random access memory, magnetic disk, optical disk) and includes several instructions to cause a terminal device (which may be a mobile phone, computer, server, air conditioner, or network device, etc.) to execute the methods described in the various embodiments of the present invention.

[0156] The above are merely preferred embodiments of the present invention and do not limit the patent scope of the present invention. Any equivalent structural or procedural transformations made based on the content of the present invention's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. A dynamic calculation method based on spectral feature coding networks, applied to the convergence and accuracy problems of high-frequency structural dynamic equations describing engineering components in engineering scenarios, characterized in that, Includes the following steps: S1: Analyze the type of structural dynamic equations and modal parameters corresponding to the dynamic problem to be solved. The structural dynamic equations include single-degree-of-freedom dynamic equations describing dynamic changes in a single dimension and partial differential equations of structural motion describing dynamic changes in spatiotemporal coupling. For the single-degree-of-freedom dynamic equation, its modal parameters and corresponding general solution structure are extracted; for the partial differential equation of motion of the structure, it is decomposed into a combination of modal vibration function and a set of time-domain single-degree-of-freedom dynamic equations through variable separation. S2: Determine the input variables of the structural dynamic equation, build a spectral feature encoding layer, and map the input variables to the spectral space related to the solution of the structural dynamic equation. The basis functions of the spectral space are determined based on the modal parameters extracted in S1. S3: Construct a neural network architecture including the spectral feature encoding layer and a multilayer perceptron, and fit the nonlinear combination relationship of the basis functions in the spectral space through the multilayer perceptron; S4: Construct a loss function based on the physical constraints of the structural dynamic equations. The loss function includes at least initial condition constraint terms and constraint terms of the equations themselves. S5: Train the neural network architecture under the constraints of the loss function, and directly output the solution of the single-degree-of-freedom dynamic equation as the result of the dynamic calculation; for the partial differential equation of structural motion, couple the time-domain solution obtained by training with the modal vibration function to obtain the final solution as the result of the dynamic calculation.

2. The dynamic calculation method based on spectral feature coding network as described in claim 1, characterized in that, The single-degree-of-freedom dynamic equation is a linear equation describing the high-frequency vibration of the engineering structure corresponding to the dynamic problem to be solved. The modal parameters of the single-degree-of-freedom dynamic equation include at least the vibration frequency and the attenuation coefficient. The general solution structure of the single-degree-of-freedom dynamic equation includes at least repeated conjugate roots.

3. The dynamic calculation method based on spectral feature coding network as described in claim 2, characterized in that, The repeated conjugate roots are expressions corresponding to the frequency and damping ratio of the engineering structure.

4. The dynamic calculation method based on spectral feature coding network as described in claim 1, characterized in that, The partial differential equation of structural motion is an equation describing multi-dimensional spatiotemporal coupled vibration. This equation is used to describe at least cable vibration. The mode shape function is an orthogonal function system that satisfies the boundary conditions. Spatiotemporal variables are decoupled by separating variables.

5. The dynamic calculation method based on spectral feature coding network as described in claim 1, characterized in that, The input variables include independent variables and equation parameters to be generalized; for high-frequency vibration problems, the input variables include at least time variables and parameters corresponding to vibration frequency and attenuation coefficient, which include at least structural frequency and damping ratio.

6. The dynamic calculation method based on spectral feature coding network as described in claim 1, characterized in that, The basis functions of the spectral feature coding layer match the general solution structure of the structural dynamic equation, and the types of the basis functions directly correspond to the modal parameters extracted in S1, including: For periodic oscillation problems, basis functions containing trigonometric functions are used; An exponential basis function is used to address the decay problem.

7. The dynamic calculation method based on spectral feature coding network as described in claim 1, characterized in that, The multilayer perceptron only performs weight learning on the basis functions in the spectral space, and achieves solution fitting through linear combination and nonlinear activation, without directly processing the original input variables.

8. The dynamic calculation method based on spectral feature coding network as described in claim 1, characterized in that, The initial condition constraint term is the error between the network output and the initial state of the dynamic problem. The initial state of the dynamic problem includes at least the initial displacement and the initial velocity. The constraint term of the equation itself is the deviation after substituting the network output into the structural dynamic equation.

9. The dynamic calculation method based on spectral feature coding network as described in claim 8, characterized in that, The initial condition constraints are balanced with the constraints of the equation itself by a preset weighting coefficient.

10. The dynamic calculation method based on spectral feature coding network as described in claim 1, characterized in that, The dynamic problem to be solved is a pre-defined engineering scenario, and the analysis of this engineering scenario includes solving the structural dynamic equations.

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