Method for solving structural dynamic response under earthquake action based on Fourier feature mapping physical information neural network

Abstract

The invention discloses a Fourier feature mapping physical information neural network-based structural dynamic response solving method under an earthquake action, which comprises the following steps of: constructing a Fourier feature neural network FF-PINN by embedding Fourier feature mapping, and solving the structural dynamic response under the earthquake action on the premise of not needing any system response data. And solving the dynamic response of the single-degree-of-freedom structural system and the two-layer shearing frame multi-degree-of-freedom structural system under the action of the actual earthquake load. The method overcomes the difficulty of a traditional PINN model in solving a target function containing high and low frequency information superposition, avoids the limitation of solving the structure displacement response under the action of seismic waves, improves the capturing capability of the PINN model for high frequency information, and achieves the accurate solving of the structure displacement response under the action of seismic waves. And from the perspective of the NTK theory, the dynamic response problem of the structure can be effectively solved when the natural vibration frequency of the structure is selected from the Fourier characteristic parameters.

Description

Technical Field

[0001] The present invention relates to a method for solving the structural dynamic response under earthquake action based on a Fourier characteristic mapping physical information neural network. Background Art

[0002] Calculating the dynamic response of structures is crucial for understanding the dynamic characteristics of engineering structures. Since the equations of motion of structures are essentially second-order differential equations, the field of structural dynamics typically uses differential equations to analyze the dynamic response of structures under external loads. Choosing an appropriate time step to ensure the stability and efficiency of the numerical solution remains a challenge in structural dynamics analysis. To overcome the limitations of traditional numerical solutions to differential equations, the application of artificial neural networks in solving differential equations has received increasing attention in recent years, becoming an emerging approach for analyzing the dynamic response of structures.

[0003] Currently, research on ANNs focuses primarily on data-driven methods and physical constraints. Data-driven methods use large amounts of data to train neural networks, learning to solve partial differential equations (PDEs) and establishing a mapping between inputs and outputs. Numerous research results have been achieved using data-driven methods in the field of structural analysis. Data-driven methods have achieved remarkable results in the prediction of structural dynamic response. However, due to their high reliance on training data, the quality of the training data must be ensured, and predictions outside the training data range often exhibit poor generalization. Physical constraints, on the other hand, do not rely on large amounts of training data. Instead, they automatically satisfy the embedded physical equations by sampling data, thus eliminating the need for large-scale data. Currently, the most widely used physical constraints method is the physical-informed neural network (PINN). This method embeds information about the PDEs into the loss function, ensuring adherence to the laws of physics during the optimization process. The introduction of physical constraints not only constrains the trainable parameters, thus addressing overfitting, but also reduces the reliance on large datasets, thereby enhancing the robustness of machine learning models. In addition to solving PDEs, this method can also effectively solve the parameters in the equations, enabling the solution of inverse problems. The PINN method provides a new approach to solving dynamic response problems and has attracted widespread attention in the field of structural engineering. However, traditional PINN models typically use a fully connected neural network architecture, which has two key weaknesses.

[0004] The first weakness is related to the varying convergence rates of different terms in the PINN loss function. For partial differential equations with high-frequency or multi-scale features, the PINN's gradient flow can become rigid during training, leading to uneven convergence rates across the loss terms and, consequently, hindering the solution process. Even when training to solve the simplest classical elliptic equations, traditional PINN models can fail.

[0005] The second weakness is the spectral bias problem. Fully connected neural networks tend to prioritize learning low-frequency information and ignore high-frequency information. This leads to poor performance of PINN when dealing with multi-frequency superposition problems. The PINN model trains and solves the one-dimensional Poisson equation, and the target solution function is in the form of a multi-frequency combination. The study found that even if the iteration exceeds 10 7 This indicates that the traditional PINN has obvious limitations in solving the objective function with multiple frequency combinations.

[0006] The limitations of fully connected neural network architectures restrict the problem-solving capabilities of the PINN model. To address this limitation, researchers currently propose optimization methods through two main approaches. The first approach is to directly replace fully connected neural networks with other types of neural network architectures, among which convolutional neural networks (CNNs) have received the most attention. Convolutional neural networks possess the invariance of symmetries and distributed pattern representation groups found in natural images, and can solve simple tasks characterized by symmetry groups. However, when scaling to more complex tasks, convolutional neural networks often require specialized design, primarily because the underlying invariances of many physical systems are difficult to explicitly encode and place high demands on the neural network architecture.

[0007] Another approach is to optimize the basic architecture of fully connected neural networks to improve their problem-solving capabilities. To address the differences in convergence rates among different terms in the PINN loss function, a learning rate annealing algorithm was proposed based on numerical experiments. This algorithm adaptively assigns appropriate weights to each term in the loss function during model training, leveraging gradient statistics to balance the convergence rates of each term. The coefficients of each term in the PINN loss function are dynamically updated based on their average convergence rate within the total training error. Summary of the Invention

[0008] To address the above technical deficiencies, the present invention provides a method for solving the structural dynamic response under earthquake action based on Fourier eigenmapping physical information neural network, which can effectively assist PINN in capturing high-frequency information and thus achieve accurate solution of the structural displacement response under earthquake action.

[0009] The present invention is achieved by the following measures:

[0010] A method for solving the structural dynamic response under earthquake action based on a Fourier eigenmap physical information neural network includes the following steps:

[0011] Step 1: The seismic wave composed of the superposition of multiple frequency fluctuations is expressed as the following formula:

[0012]

[0013] Among them, ω n is the nth frequency component, A n 、B n 、 are the amplitude and phase of the frequency component respectively; then, based on this formula, the Fourier map is embedded to construct the Fourier feature neural network model; the Fourier feature map is defined as:

[0014]

[0015] Step 2: The forward propagation of the Fourier feature neural network model through the fully connected layer is defined as follows:

[0016] H1=φ(W1γ(t)+b1) (3.2)

[0017] H l =φ(W l H l-1 +b l ),l=2,...,L (3.3)

[0018] u θ (t) = W L+1 H L +b L+1 (3.4)

[0019] Where φ, W, and b are the activation function, weight, and bias of the neural network respectively, and B=[a1,a2,...,a m ] T Sampling from Gaussian distribution N(o,σ 2 ), whose sampling value remains fixed during the training process. The choice of Gaussian distribution variance, that is, Fourier characteristic parameter σ, is related to the frequency of the target function to be solved. Each characteristic parameter σ i The corresponding Fourier feature output is And combine multiple feature outputs, and then output them as the objective function through the final linear layer:

[0020]

[0021] Then, the dynamic responses of the single-degree-of-freedom structural system and the two-story shear frame multi-degree-of-freedom structural system under actual earthquake loads are solved according to the above formula;

[0022] Step 3: Finally, numerical experimental analysis is performed using the NTK theory to determine the influence of characteristic parameters on the frequency range of the learning objective function of the Fourier characteristic neural network model.

[0023] Based on the NTK theory, a neural network with Fourier characteristics of a hidden layer:

[0024]

[0025] Where t is the input sampled at time, W is the weight matrix, and the NTK matrix K of the neural network ij It can be expressed as:

[0026]

[0027] The eigenvalues ​​and corresponding eigenfunctions of the NTK matrix can be expressed as follows:

[0028]

[0029] g(x)=C1 cos(at)+C2 sin(at)(3.9)

[0030] Where C1 and C2 are constants. When the value of σ is larger, the value distribution width of the sampled characteristic point a is larger, the NTK matrix presents a higher-frequency characteristic function, and the distribution of the eigenvalues ​​will be tighter. In structural dynamics problems, the response function of the structure to external loads is usually related to the natural frequency of the structure, that is, there are terms related to the natural frequency in the frequency of the response function. When the characteristic parameter is taken as the natural frequency, the characteristic vector is obtained by sampling the characteristic points with the natural frequency as the variance as follows:

[0031]

[0032] The above formula gives the expression of circular frequency and the conversion relationship with frequency, f n is the frequency.

[0033] When σ=10, the model has the best solution effect, the error is reduced to 1.06%, and the model can reach convergence stability faster; when the iteration is 6×10 3 When the L2 norm error is about 5%, the FF-PINN model has excellent solution efficiency and accuracy under this parameter.

[0034] The beneficial effects of the present invention are as follows: By embedding Fourier eigenmaps and constructing a Fourier eigenneural network (FF-PINN), this patent solves the dynamic response of single-degree-of-freedom structural systems and two-layer shear frame multi-degree-of-freedom structural systems under actual seismic loads without requiring any system response data. This overcomes the difficulties of the traditional PINN model in solving objective functions that contain superpositions of high- and low-frequency information, avoids the limitations of solving structural displacement responses under seismic waves, and enhances the PINN model's ability to capture high-frequency information, thereby achieving an accurate solution to the structural displacement response under seismic motion. Furthermore, from the perspective of NTK theory, it is demonstrated that the dynamic response of a structure can be effectively solved when the Fourier eigenvalue parameters are selected as the natural frequency of the structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 Flowchart of the PINN model for solving the structural dynamics;

[0036] Figure 2 Diagram of the structural dynamic response solution architecture based on the FF-PINN model;

[0037] Figure 3 Schematic diagram of the displacement response and L2 norm error of the FF-PINN model under the excitation force p(t) = 10sin(4t);

[0038] Figure 4 Schematic diagram of the FF-PINN displacement solution and L2 norm error under the exciting force p(t) = sin(t) + 15cos(50t);

[0039] Figure 5 Schematic diagram of the mechanical model of a two-layer shear frame structure;

[0040] Figure 6 Schematic diagram of the displacement response and L2 error (second and first floors, respectively) of a two-story frame structure subjected to earthquake excitation;

[0041] Figure 7 Schematic diagram of the L2 norm error of the σ(2, 5, 10, 20, 50) FF-PINN model;

[0042] Figure 8 Spectral analysis diagram of the displacement response of the structural system under the action of earthquake motion EQ2;

[0043] Figure 9 Schematic diagram of the spectrum analysis of the maximum eigenvalue eigenvector of the neural network NTK matrix under different σ (2, 5, 10, 20, 50); DETAILED DESCRIPTION

[0044] The present invention will be described in further detail below with reference to the accompanying drawings:

[0045] Using the Physical Information Neural Network (PINN) modeling method, a traditional PINN model for solving structural dynamic responses was constructed. This model was used to calculate the dynamic response of a single-degree-of-freedom structural system under a single-frequency simple harmonic load and a combination of two-frequency simple harmonic loads. The partial differential equation defined as follows was used:

[0046] L[u](x)=f(x),x∈Ω (1.1)

[0047]

[0048] Where Ω is the solution domain, is the solution boundary, L is the differential operator, and u(x) is the solution to the partial differential equation. Sample x from the solution domain and the solution boundary, and output u(x,θ) through the fully connected neural network forward propagation, where θ is the neural network parameters (including weight parameters and bias). The forward propagation process of the fully connected network is as follows:

[0049] The input layer is defined as

[0050] f (0) (x)=x (1.3)

[0051] For input dimension d o =d, output dimension d L+1 = 1, the forward propagation of a fully connected neural network with L hidden layers is

[0052]

[0053] g (h) (x)=φ(W (h-1) f (h-1) (x)+b (h-1) )(1.5)

[0054] Where W is the weight of the network, φ is the activation function of the network, and d h is the output dimension of the hth hidden layer, g (h) (x) is the network output after the activation function and is passed to the next layer of neural network to obtain f (h) (x).

[0055] The final output

[0056]

[0057] The neural network backpropagates through the loss function to update the network parameters. The residual of this partial differential equation is defined as:

[0058] r(x,θ)=Lu(x,θ)-f(x)(1.7)

[0059] Therefore, the loss function of the model can be expressed as

[0060] L(θ)=L b (θ)+L r (θ)(1.8)

[0061] Among them, L b Represents the boundary condition loss term of the equation, L r Represents the loss term of the differential equation. The specific expressions of the loss function are as follows:

[0062]

[0063] where N b With N r is the number of sampling points on the boundary and inside the equation The general process of the PINN model is discussed above. For structural dynamics, the differential equation term of formula (1.1) is

[0064]

[0065] Where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the system, respectively. t is time, and U is the displacement vector of the structure. Solving this differential equation requires initial boundary conditions. In structural dynamics, the displacement and velocity conditions of the structure at the initial moment are usually used to solve it.

[0066]

[0067] Therefore, in the constructed structural dynamics PINN model, the loss function is as follows:

[0068]

[0069] The total loss is as shown in formula (1.4), which includes the equation loss term and the boundary loss term.

[0070] Based on the PINN principle, a solution model for a single degree of freedom system in structural dynamics was constructed and two examples were solved. The motion equations for a single degree of freedom system are:

[0071]

[0072] Where m, c, and k are the mass, damping, and stiffness of the system, respectively; t is time; and u is the displacement of the structure. In the example set in this section, the initial velocity and displacement are both 0. This means there are two boundary conditions:

[0073]

[0074] Therefore, in the constructed structural dynamics PINN model, the loss function is as follows

[0075]

[0076] First, the traditional PINN model is used to solve the displacement response of an undamped system subjected to a simple harmonic load. The mass of the structure is 1kg, the stiffness is 36N / m, and the time domain t∈[0,10s]. The L2 norm error of formula (1.19) is used to evaluate the solving ability of the trained model. The smaller the L2 norm error, the stronger the model fitting ability. are the displacement prediction value of the model and the true solution of the differential equation of motion, respectively. The true solution for a single-degree-of-freedom undamped system subjected to a simple harmonic load can be obtained analytically, but analytical solutions for more complex problems are not available. In this case, the displacement response obtained using the Newmark method with a small step size of 0.01s is used as a reference solution.

[0077]

[0078] The solution settings of the PINN model are as follows: the number of sampling points inside the equation is 1000, the number of sampling points at the boundary points is 400, the learning rate is set to 0.001, the number of hidden layers of the neural network is 4, the number of neurons in each layer is 100, the activation function is the tanh function, the optimizer is the Adam algorithm, and a 5×10 4 First, the displacement response of a simple harmonic load with a single frequency of p(t)=10sin(4t) was solved. Based on the above neural network model settings, the external load was changed to a simple harmonic load with a high and low frequency combination of p(t)=sin(t)+15cos(50t). The solution was obtained in the PINN model training iteration 1×10 6 The result of times.

[0079] When the structure is subjected to a simple harmonic load of a single frequency, the PINN model can be trained to obtain the displacement response of the problem, with the L2 norm error reduced to 4.06%. This indicates low accuracy for the problem being solved. Although increasing the number of sampling points or iterations may improve the solution, this reduces solution efficiency and consumes more computing resources. The PINN model often faces a trade-off between accuracy and efficiency, which limits its application in practical engineering.

[0080] Furthermore, the PINN model can only capture the low-frequency portion of the displacement response and cannot effectively fit the high-frequency components. However, the loads borne by real structures are often complex and random. For example, earthquake waves are a type of load with multiple frequencies superimposed on them. They can be approximately expressed as a superposition of simple harmonic waves (sine and cosine waves) of various frequencies and amplitudes. Equation (1.20) illustrates the process of analyzing the spectrum of a discrete signal.

[0081]

[0082] The above formula performs a discrete Fourier transform on the input signal and calculates the amplitude, where X[k] is the complex spectrum of the input signal x[n] after Fourier transform, N is the number of sampling points, k represents the frequency index, Re(X[k]) and Im(X[k]) represent the real and imaginary parts of X[k] respectively. If a single-sided spectrum is drawn, as shown in the following formula

[0083]

[0084] The frequencies after discrete Fourier transform are:

[0085]

[0086] where f s is the sampling frequency.

[0087] Analysis of the acceleration time history and Fourier spectrum of the 1940 El Centro earthquake shows that earthquake waves are composed of a combination of fluctuations at multiple frequencies, as expressed in Equation (1.23). The PINN solution above demonstrates that this model is very difficult to use in determining the dynamic response of structures subjected to earthquake waves.

[0088]

[0089] Among them, ω n is the nth frequency component, A n 、B n 、 are the amplitude and phase of the frequency component respectively.

[0090] The traditional PINN model has difficulties in solving the objective function that contains the superposition of high- and low-frequency information, which limits its ability to solve the displacement response of the structure under the action of seismic waves. In order to solve this problem and improve the PINN model's ability to capture high-frequency information, this patent introduces Fourier mapping to construct a Fourier characteristic neural network FF-PINN. This model can effectively assist PINN in capturing high-frequency information, thereby achieving an accurate solution to the displacement response of the structure under seismic motion. In addition, this patent will continue to explain the reasons for the spectral deviation phenomenon of the traditional PINN model from the perspective of NTK theory, and the mechanism of alleviating this phenomenon by embedding Fourier mapping. At the same time, it explains that when the Fourier characteristic parameters are selected as the natural frequency of the structure, the dynamic response problem of the structure can be effectively solved.

[0091] Through numerical experiments, it is found that the spectral bias weakness of the PINN model using a fully connected neural network restricts the solution capability of this method. The neural tangent kernel is defined as follows when the number of iterations t and the network parameter θ(t) are:

[0092]

[0093] When the width of the hidden layer tends to infinity, the kernel function converges to a deterministic kernel with probability when randomly initialized. This property can help understand some behaviors of neural networks during training, such as the spectral deviation behavior of the PINN model, which is a fully connected neural network. The neural tangent kernel matrix (NTK) of the traditional PINN model. Consider minimizing the loss function (1.8) by gradient descent and a very small learning rate, thereby generating a gradient flow

[0094]

[0095] make Then give the sampling point And the gradient flow (2.2), u(t), L(u(t)) follows the following evolution law

[0096]

[0097] where K ru (t) = K T ur (t), and there is

[0098]

[0099] where <·,·> is defined as the inner product of all parameters θ, as follows

[0100]

[0101] For K uu (t), K rr (t) and K(t) are both semi-positive matrices, if we define J u (t) and J r (t) is the Jacobian matrix of u(t), Lu(t) for parameter θ, then

[0102]

[0103] The above describes the calculation method of the NTK matrix of a general PINN. For a fully connected neural network with only one hidden layer and sufficient width, it is proved that K(t)≈K(0), That is, the NTK matrix of the neural network remains stable during the iteration process, so Equation (2.2) can be written as

[0104]

[0105] Where K(0) = Q T ΛQ, Q, Λ are orthogonal matrices and diagonal matrices respectively, and the value of Λ is determined by the eigenvalue λ of K(0) i Composition, so we can get

[0106]

[0107] This means that the i-th component on the left side of Equation (2.8) will be approximately The rate of decay is . In other words, the eigenvalues ​​of the neural tangent kernel matrix describe the rate at which the absolute training error decreases, and the objective function components corresponding to eigenvectors with larger eigenvalues ​​will be learned faster. For fully connected networks, eigenvectors with higher eigenvalues ​​of the NTK matrix usually exhibit lower frequencies, and the eigenvalues ​​of NTK decay quickly. In other words, the eigenvectors corresponding to larger eigenvalues ​​dominate the convergence direction of the neural network, resulting in extremely slow convergence to the high-frequency components of the objective function. Therefore, the PINN model using a fully connected network will suffer from spectral bias.

[0108] The reason for the spectral deviation phenomenon of the PINN model under the fully connected neural network is given from the neural tangent kernel theory. From this perspective, the NTK matrix of the set PINN model is solved through numerical experiments. The eigenvectors corresponding to the larger eigenvalues ​​of the NTK matrix contain lower frequency information. As the eigenvalue decays, the frequency increases accordingly. Because the decay rate of the eigenvalue is very fast, the convergence direction of the neural network is dominated by the eigenvectors corresponding to the large eigenvalues, that is, it converges preferentially to the low-frequency direction.

[0109] By embedding the Fourier map, the Fourier characteristic neural network model FF-PINN of structural dynamics is constructed. The displacement response u(t) of the structure can be expressed as Equation (1.23) through Fourier decomposition similar to the seismic wave. Therefore, a Fourier characteristic neural network model is constructed. The network model outputs a series of frequency components and combines them into the final displacement response result through a linear layer. Obviously, the Fourier characteristic map can meet this requirement. For the time-dependent differential equation of motion, the input time information can be mapped to the Fourier function space, and then input into the fully connected layer for further processing and solution. The Fourier characteristic map is defined as:

[0110]

[0111] The forward propagation through the fully connected layer is defined as follows:

[0112] H1=φ(W1γ(t)+b1) (3.2)

[0113] H l =φ(W l H l-1 +b l ),l=2,...,L (3.3)

[0114] u θ (t) = W L+1 H L +b L+1 (3.4)

[0115] Where φ, W, and b are the activation function, weight, and bias of the neural network respectively, and B=[a1,a2,...,a m ] T Sampling from Gaussian distribution N(o,σ 2 ), whose sampling value remains fixed during the training process. The choice of Gaussian distribution variance, namely Fourier characteristic parameter σ, is generally related to the frequency of the target function to be solved. Common selection values ​​include 1, 20, 50, etc. Fourier mapping can effectively learn high-frequency information and improve the effectiveness of neural networks in solving various practical problems. In order to ensure that the Fourier characteristic neural network can accurately learn multiple frequency information, multiple Fourier characteristic mapping layers can be used. The neural network architecture for solving structural dynamic response is as follows: Figure 2 As shown, each characteristic parameter σ i The corresponding Fourier feature output is obtained as (3.4) Combine multiple feature outputs and output them through the final linear layer as the target function:

[0116]

[0117] This section continues to explain the mechanism by which Wang et al.

[32] use the NTK theory to explain how Fourier feature maps assist fully connected neural networks in learning high-frequency information. Consider a neural network with only one hidden layer and Fourier features:

[0118]

[0119] Where t is the input sampled at time, W is the weight matrix, and the NTK matrix K of the neural network ij It can be expressed as:

[0120]

[0121] The eigenvalues ​​and corresponding eigenfunctions of the NTK matrix can be expressed as follows:

[0122]

[0123] g(x)=C1 cos(at)+C2 sin(at)(3.9)

[0124] Where C1 and C2 are constants. By analyzing the NTK matrix of the Fourier characteristic network, it can be found that when the value of σ is larger, the value distribution width of the sampled characteristic point a will be larger, the NTK matrix will show a higher frequency characteristic function, and the distribution of the characteristic value will be tighter. Therefore, the Fourier characteristic mapping can effectively solve the spectral deviation problem of PINN. In structural dynamics problems, the response function of the structure to external loads is usually related to the natural frequency of the structure, that is, there are terms related to the natural frequency in the frequency of the response function. When the characteristic parameter is taken as the natural frequency, by sampling the characteristic points with the natural frequency as the variance, the characteristic vector such as (3.10) will be obtained. According to the above derivation, it can be seen that the neural network will move towards the frequency ω n , thereby alleviating the problem that traditional PINN cannot capture high-frequency information and helping the neural network to solve the dynamic response of the target structure faster.

[0125]

[0126] The above formula gives the expression of circular frequency and the conversion relationship with frequency, f n is the frequency.

[0127] First, two examples are calculated using the FF-PINN model. The natural frequency is 6 rad / s, so the Fourier characteristic parameter σ is selected as 6. The results are shown in Figure 2. Figure 3 and Figure 4 As shown. Figure 3 It can be seen that compared with the traditional PINN model, FF-PINN has a faster convergence speed and higher solution accuracy. In this problem, the L2 norm error is reduced to below 1%, showing excellent solution accuracy. Figure 4 The results show that FF-PINN has the ability to solve the problem of high- and low-frequency superposition objective functions, and its L2 norm error is reduced to 4.46%. Compared with the solution results of the traditional PINN model, the introduction of Fourier eigenmap significantly improves the solution ability of PINN.

[0128] In summary, the Fourier map effectively assists the fully connected neural network (PINN) architecture in capturing high-frequency information, thereby improving the model's practicality. For structural dynamics problems, it provides a new approach for solving structural responses to multi-frequency superimposed loads such as earthquake motion. Next, this section will explore the performance of the FF-PINN model in analyzing structural responses to earthquake motion.

[0129] Single degree of freedom structure solution example:

[0130] First, a single-degree-of-freedom (DOF) viscous damping system was set up under seismic excitation, and the FF-PINN model was used to solve the displacement response under this condition. To verify the solving ability of the neural network model, the calculation results of the FF-PINN model were compared with the displacement response results solved using the Newmark method. The seismic acceleration data are discrete values ​​that depend on time. The selected seismic data are 11 data points selected from the seismic records recommended by FEMAP695, as shown in Table 1. Interpolation calculations were performed on these discrete data points so that they could be used as the input P(t) in the equation. Tables 2 and 3 show the model parameter settings for the single-degree-of-freedom system and the basic parameters of the neural network. The natural frequency of the structure is approximately 14 rad / s, so the Fourier characteristic parameter σ is set to 14, and 50 characteristic points are sampled within the Gaussian distribution.

[0131] Table 1: Selected seismic information

[0132]

[0133] Table 2: Parameters of single degree of freedom system

[0134]

[0135] Table 3: Basic parameters of neural network

[0136]

[0137] Under the above parameter settings, it is possible to train for 2×10 iterations without any system response data. 4 In the first step, the L2 norm error is used to evaluate the model’s fitting ability. The traditional PINN cannot solve the corresponding problem of the structure under the action of earthquake motion, and its L2 norm error is as high as 93%, while the calculation results of FF-PINN are roughly consistent with the numerical solution. 4 When the FF-PINN model begins to converge, its L2 norm error is approximately 6%, demonstrating excellent solution efficiency. Therefore, introducing Fourier eigenmaps into the traditional PINN network to decompose time scale information not only solves problems that traditional PINN models find difficult to handle, but also performs well in terms of solution efficiency.

[0138] Table 4: L2 norm error of 11 earthquake displacement responses solved by FF-PINN

[0139]

[0140] Table 4 shows the L2-norm error of the displacement responses of 11 earthquake motions solved using the aforementioned FF-PINN model. The error level is consistently reduced to less than 4%. This demonstrates that Fourier mapping, using the structural natural frequency as a characteristic parameter, delivers excellent solution efficiency and accuracy. These examples demonstrate the significant potential of FF-PINN for solving structural responses to earthquake motions. By embedding Fourier mapping within a fully connected neural network architecture, FF-PINN fully captures the periodic nature of the displacement response and rapidly approximates the true solution.

[0141] Two-layer shear frame structure solution example:

[0142] The FF-PINN model was used to solve the dynamic response of a single-degree-of-freedom system subjected to seismic loads, verifying its solving capabilities. Actual engineering structures in structural dynamics often have multiple degrees of freedom. The mathematical models are shown in Equations (1.11) to (1.14). The FF-PINN model was used to solve the displacement response of a two-degree-of-freedom model subjected to the El Centro earthquake. By employing two Fourier eigenmaps, a model based on two eigenparameter mappings was constructed. Figure 5 The mechanical model of the two-layer shear frame structure is as follows: the concentrated mass of each layer is m = m1 = m2 = 25000 kg, and the stiffness EI is 26.666 N·m 2 , floor height h = 2.8m, Rayleigh damping coefficient α = 0.5, β = 0.002, the stiffness matrix, mass matrix, and damping matrix of the structure are calculated as follows:

[0143]

[0144] The main frequencies of the two degrees of freedom of the structure are calculated to be approximately 14 rad / s and 39 rad / s, which are selected as characteristic parameters for mapping. The time interval of the selected seismic motion record is 0.02 seconds, and the theoretical solution of the seismic response of the multi-degree-of-freedom system cannot usually be directly calculated. In order to obtain reasonable and accurate results, the time step of the Newmark method is defined as 0.005 seconds, which requires the interpolation of the seismic motion record. After interpolation, the dynamic response calculated by the Newmark method is used as a benchmark and compared with the calculation results of the proposed method to verify the effectiveness of the proposed method. Figure 6 As can be seen, the displacement response calculated by the FF-PINN model is highly consistent with the results of the traditional Newmark method at a step size of 0.005 seconds. Therefore, the PINN model cannot solve the problem. The FF-PINN model constructed in this section demonstrates excellent performance in solving the dynamic response problem of multi-degree-of-freedom systems.

[0145] Selection of Neural Network Hyperparameters

[0146] When training a neural network, choosing appropriate hyperparameters is crucial for the model's convergence speed, training stability, and ultimate performance. Proper hyperparameter configuration can significantly improve the model's accuracy and generalization. Therefore, a thorough understanding of the impact of hyperparameters on model performance and optimizing them are essential steps in training a neural network. This section explores the impact of hyperparameters on the FF-PINN model.

[0147] The learning rate determines the magnitude of the model weight update. A too high learning rate may lead to unstable training or even divergence; while a too low learning rate may make the training speed too slow, reduce the efficiency of the model solution, or even lead to falling into a local optimal solution and being unable to effectively solve the problem. Based on the example of a single-degree-of-freedom structural system, the influence of the learning rate on the convergence properties of the solution is discussed by adjusting the learning rate and comparing the L2 norm error. This paper selects four cases with learning rates of 0.01, 0.001, 0.0001 and 0.00001 respectively. When the learning rate is 0.01, it is too large, resulting in the model being unable to converge effectively and the solution being unstable; when the learning rate is 0.00001, although increasing the number of iterations may eventually get a solution, the solution efficiency is poor. When the learning rates are 0.001 and 0.0001, the model shows good efficiency and accuracy. Among them, when the learning rate is 0.001, its convergence speed is faster, and its iteration to 5×10 3 times and iterated to 1.25×10 at a learning rate of 0.0001 4 The L2 error level is comparable.

[0148] The depth and width of a neural network are determined by the number of hidden layers and the number of neurons in each layer. The depth and width of the network directly impact the neural network's ability to fit the problem. For specific problems, a complex neural network may lead to overfitting and reduce computational efficiency, while an overly narrow neural network may underfit. Therefore, selecting the appropriate neural network depth and width is crucial to improving the model's ability to solve problems. The impact of neural network depth and width on the fitting ability of the FF-PINN model is discussed. Increasing the number of neurons can increase the convergence rate of the solution, but increasing the number beyond a certain point slows down the convergence rate, thereby reducing computational efficiency. In the experiments discussed, the model with 70 neurons achieved the best fitting accuracy. The impact of the number of hidden layers on convergence speed is similar to that of the number of neurons. Excessive hidden layers do not significantly improve solution accuracy, but instead reduce convergence speed. In the experiments discussed, the model with three hidden layers achieved the best fitting efficiency.

[0149] The Fourier characteristic parameter σ represents the variance of the Gaussian distribution. Adjusting the size of this parameter can change the number of Fourier transform features. Generally, as the value of σ increases, the range of the Fourier characteristics will also expand, which will help the fully connected neural network fit the multi-frequency objective function problem. The solving ability of the FF-PINN model under different σ parameter values. The basic parameters of the neural network are shown in Table 3. The natural frequency of the structure is approximately 14.14 rad / s (2.25 Hz), the seismic motion is EQ2, and the number of iterations is set to 2×10 4 The σ parameter was selected as 2, 5, 10, 20 and 50 respectively.

[0150] Figure 7 The L2 norm error of the solution results of the FF-PINN model and the Newmark method under the above five parameters is shown. First, it can be seen from the figure that, except when σ=50, the model fails to complete the solution task within the given number of iterations, and its L2 norm error is 98.90%, and the fitting effect is very unsatisfactory; under the other four conditions, the L2 norm error can be reduced to below 10%. Among them, when σ=10, the solution effect of the model is the best, and the error is reduced to 1.06%, which is the lowest among the five parameters. Secondly, from Figure 9 It can also be found that when σ = 10, the model can reach convergence stability faster than other cases. Through multiple independent experiments, it was found that when the iteration is 6×10 3 When the L2 norm error is about 5%, the FF-PINN model has excellent solution efficiency and accuracy under this parameter.

[0151] From the perspective of NTK theory, we rigorously derived the effect of the parameter σ on fully connected neural networks under the condition of a single-layer neural network. However, practical applications typically involve neural networks with multiple hidden layers, and analytical analysis of the NTK matrix for these layers is difficult. This section will continue with numerical experiments, attempting to explore the reasons for the differences in neural network model solving capabilities under different values ​​of σ by analyzing the eigenvectors of the NTK matrix. Figure 8 The displacement spectrum analysis diagram of the target structure system shows that its frequency information is concentrated at 2.21Hz. Figure 9 The eigenvector spectrum analysis diagram corresponding to the maximum eigenvalue of the NTK matrix under different initial states of the experiment is shown. The response of a structure under earthquake motion is usually closely related to its natural frequency. The natural frequency of this structure is about 2.25Hz. It can be seen that the target displacement function of this example is indeed highly correlated with the natural frequency of the structure. Figure 9 It can be seen that as the parameter σ increases, the frequency component width of the characteristic function also becomes wider, and the ability of the neural network to capture high-frequency information is enhanced. Figure 9 (b) (c) correspond to parameters 5 and 10. The neural network's solving ability under these parameters is Figure 8 It can be seen that it is better than other parameters. From the frequency information decomposed in its spectrum diagram, it can be seen that the main reason may be that the spectrum information preferentially captured by the neural network affected by it is close to the spectrum information of the target displacement function. Figure 9 Although the spectral analysis in (d) and (e) affects the 2.21Hz frequency component, it decomposes too much high-frequency information, resulting in a decrease in the neural network's solution efficiency. The above analysis shows that a solution can still be achieved when using relatively small characteristic parameters, such as when σ = 2 in this example, but both the solution efficiency and accuracy will be reduced. Efficiency improves when the frequency information of the eigenvector corresponding to the maximum eigenvalue of the neural network's NTK matrix under the selected parameters is close to the target function. However, when the frequency information decomposed by excessively large parameters deviates significantly from the target function, the solution efficiency is too low. Combining the above analysis with the numerical example results in this paper, it is often appropriate to select the natural frequency of the structure as the characteristic parameter. For the problem of the response of structural systems to earthquake motion, combining Fourier eigenmaps with fully connected neural networks can successfully solve the displacement response of structural systems under earthquake motion without requiring any system response data. The main conclusions are as follows: Fourier eigenmaps can assist fully connected neural networks in enhancing their ability to capture high-frequency information, enabling them to solve the dynamic response of structures under seismic motion. They are efficient, with an L2 error of less than 4% compared to the Newmark method. Selecting appropriate eigenmap parameters can further reduce this error to less than 2% or even lower. Frequency-domain analysis of the eigenvectors of the NTK matrix reveals that a larger Fourier eigenvalue parameter, σ, increases the frequency of the target function it can learn, enabling it to solve structural response problems with multiple frequencies superimposed. It was also found that a larger eigenvalue parameter results in more decomposed frequencies. When the decomposed frequencies deviate more from the target function frequencies, the model's solver performance is reduced. When the decomposed frequencies are too few to encompass the target function frequencies, the model's solver efficiency is also reduced. Selecting the structural natural frequency as the eigenvalue parameter often results in better solver performance.

[0152] In summary, this paper proposes a new method for solving the dynamic response of structures under earthquake motion. By using Fourier eigenmaps to assist PINN learning, this method can effectively and accurately solve the dynamic response of structures under earthquake motion without requiring any system response data. When solving multi-degree-of-freedom structural systems, this paper uses the natural frequency of the structure as the eigenvalue parameter of the Fourier map, obtaining results that are highly consistent with traditional numerical algorithms. However, it is important to note that the randomness of the eigenvalues ​​sampled from the Gaussian distribution can affect the neural network's solution process.

[0153] The above is only a preferred embodiment of this patent. It should be pointed out that for ordinary technicians in this technical field, several improvements and substitutions can be made without departing from the technical principles of this patent. These improvements and substitutions should also be regarded as the scope of protection of this patent.

Claims

1. A method for solving the structural dynamic response under earthquake action based on Fourier eigenmap physical information neural network, characterized in that: The following steps are involved: Step 1: The seismic wave composed of the superposition of multiple frequency fluctuations is expressed as the following formula: Among them, ω n is the nth frequency component, A n 、B n 、 are the amplitude and phase of the frequency component respectively; then, based on this formula, the Fourier map is embedded to construct the Fourier feature neural network model; the Fourier feature map is defined as: Step 2: The forward propagation of the Fourier feature neural network model through the fully connected layer is defined as follows: H1=φ(W1γ(t)+b1) (3.2) H l =φ(W l H l-1 +b l ),l=2,...,L (3.3) u θ (t)=W L+1 H L +b L+1 (3.4) Where φ, W, and b are the activation function, weight, and bias of the neural network respectively, and B=[a1,a2,...,a m ] T Sampling from Gaussian distribution N(o,σ 2 ), whose sampling value remains fixed during the training process. The choice of Gaussian distribution variance, that is, Fourier characteristic parameter σ, is related to the frequency of the target function to be solved. Each characteristic parameter σ i The corresponding Fourier feature output is And combine multiple feature outputs, and then output them as the objective function through the final linear layer: Then, the dynamic responses of the single-degree-of-freedom structural system and the two-story shear frame multi-degree-of-freedom structural system under actual earthquake loads are solved according to the above formula; Step 3: Finally, numerical experimental analysis is performed using the NTK theory to determine the influence of characteristic parameters on the frequency range of the learning objective function of the Fourier characteristic neural network model.

2. The method for solving the structural dynamic response under earthquake action based on the Fourier eigenmap physical information neural network according to claim 1 is characterized by: Based on the NTK theory, a neural network with Fourier characteristics of a hidden layer: Where t is the input sampled at time, W is the weight matrix, and the NTK matrix K of the neural network ij It can be expressed as: The eigenvalues ​​and corresponding eigenfunctions of the NTK matrix can be expressed as follows: g(x)=C1cos(at)+C2sin(at)(3.9) Where C1 and C2 are constants. When the value of σ is larger, the value distribution width of the sampled characteristic point a is larger, the NTK matrix presents a higher-frequency characteristic function, and the distribution of the eigenvalues ​​will be tighter. In structural dynamics problems, the response function of the structure to external loads is usually related to the natural frequency of the structure, that is, there are terms related to the natural frequency in the frequency of the response function. When the characteristic parameter is taken as the natural frequency, the characteristic vector is obtained by sampling the characteristic points with the natural frequency as the variance as follows: The above formula gives the expression of circular frequency and the conversion relationship with frequency, f n is the frequency.

3. The method for solving the structural dynamic response under earthquake action based on the Fourier eigenmap physical information neural network according to claim 2 is characterized by: When σ=10, the model has the best solution effect, the error is reduced to 1.06%, and the model can reach convergence stability faster; when the iteration is 6×10 3 When the L2 norm error is about 5%, the FF-PINN model has excellent solution efficiency and accuracy under this parameter.

Images