A structure response prediction method of fusing physical information with LSTM

Abstract

The application belongs to the cross field of civil structure engineering and computer science, and discloses a structure response prediction method fusing physical information, steps comprising: step 1.1, generating fluctuating wind speed according to an AR model; step 1.2, generating average wind speed by adopting average wind speed index law; step 1.3, adding fluctuating wind speed and average wind speed to obtain total wind speed, converting the total wind speed into wind pressure according to Bernoulli equation, and equivalently into concentrated force at the mass point; step 1.4, calculating the structure response of the equivalent concentrated force by adopting the average acceleration method in the Newmark-beta method; step 1.5, taking the concentrated force as input and the structure response of each mass point as output to generate training samples and test samples; step 2, constructing and training LSTM; step 3, testing the prediction accuracy and generalization ability of the trained LSTM by adopting test samples; step 4, inputting the load into the trained LSTM, and the LSTM outputs the predicted displacement, velocity and acceleration.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of civil engineering and computer science, and specifically relates to an LSTM method for predicting structural response that integrates physical information. Background Technology

[0002] Currently, numerical methods for calculating the wind-induced structural response based on the finite element model have drawbacks. The finite element method involves long modeling times and computationally intensive solutions with small time steps. Furthermore, considering the uncertainties involved in wind engineering, a large number of responses are required to obtain necessary statistical information, which continuously increases computational costs.

[0003] Artificial neural networks (ANNs) are an efficient method for calculating structural responses under wind loads, offering advantages such as short computation time and no need for finite element modeling. Currently, based on whether prior formulas are included in the loss function construction, neural networks are divided into two modes: purely data-driven and physics-integrated. Purely data-driven neural networks are suitable for applications with a large number of observations, but their disadvantage lies in the potential for poor model generalization performance due to observation bias. Furthermore, since purely data-driven neural networks do not consider whether the data patterns conform to the physical laws of the phenomena, the model's inferences based on the data may show discrepancies between data patterns and physical laws. Embedding physical equations into the neural network is one way to address this problem. Physical-informed Neural Networks (PINNs) are not opposed to purely data-driven neural networks; rather, they represent a dual-driven approach that combines the advantages of both data-driven and traditional physical equation-based methods—a "physical" and "data"-driven approach.

[0004] Existing Long Short-Term Memory (LSTM) neural network prediction methods based on structural dynamic equations only perform single-dimensional predictions of the displacement of a single mass point. They also use numerical differentiation to calculate the velocity and acceleration required by the embedded structural dynamic equations for this displacement, thus constructing a structural dynamic equation loss function for the predicted value. However, this method can only predict a single dimension, and the accuracy of the numerical differentiation calculation is relatively low. Therefore, the key problems to be solved in LSTM currently lie in:

[0005] 1. How to achieve multi-dimensional prediction, that is, to simultaneously predict the displacement, velocity, and acceleration response of all mass points in the structure;

[0006] 2. How to avoid using numerical differentiation to construct the loss function of the structural dynamic equations. Summary of the Invention

[0007] To overcome the shortcomings of the prior art, this invention provides an LSTM prediction method for structural response that integrates physical information. The method uses LSTM to predict the response of all structures under wind load in multiple dimensions, including the displacement, velocity and acceleration of each mass point. This avoids the need for low-precision numerical differentiation of displacement while meeting the requirements of multi-dimensional prediction.

[0008] This invention provides the following technical solution: an LSTM method for predicting structural response by fusing physical information, comprising the following steps:

[0009] Step 1: Generate training and test samples based on the AR model and the Newmark-β method;

[0010] Step 2: Build and train the LSTM, and set the number of iterations to 1000 to bring the LSTM loss function to converge.

[0011] Step 3: Use test samples to test the prediction accuracy and generalization ability of the trained LSTM to ensure that the trained LSTM has sufficient prediction accuracy and the same accuracy for all samples.

[0012] Step 4: Input the load into the trained LSTM, and the LSTM will output the predicted displacement, velocity, and acceleration.

[0013] Preferably, the specific steps for generating training and testing samples in step 1 include:

[0014] Step 1.1: Generate fluctuating wind speed based on the AR model;

[0015] The method for generating fluctuating wind speed using AR models is as follows:

[0016] The AR model of the column vector V(X,Y,Z,t) of spatially correlated fluctuating wind speeds at M points can be represented as:

[0017]

[0018] In the formula, X = [x1, ..., x M ] T Y = [y1, ..., y M ] T Z = [z1, ..., z M ] T , (x i ,y i ,z i ) represents the coordinates of the i-th point in space, i = 1, ..., M; p is the order of the AR model; t is time; Δt is the time step of the simulated wind speed time history; φ k Let N(t) be the autoregressive coefficient matrix of the AR model, which is an M×M square matrix, k = 1, ..., p; N(t) is the vector of independent stochastic processes.

[0019] N(t) = L·n(t) (2)

[0021] In the formula, n(t) = [n1(t), ..., n M (t)] T n i (t) is a normally distributed random process with mean 0, variance 1, and all variables are independent, i = 1, ..., M; L is an M-order lower triangular matrix, and the variables are covariance matrix R. N The Cholesky decomposition determines:

[0022] R N =L·L T (3)

[0024] For simplicity, V(X, Y, Z, t) is abbreviated to V(t). Based on the wind speed time history assumption, both sides of formula (1) are multiplied by V on the right. T (t-jΔt), we get:

[0025]

[0026] In the formula, j = 0, ..., p is used for mathematical expectation operation (Expectation[]) and combined with the following properties of the autocorrelation function:

[0027] R(-jΔt)=E[V(t)V T (t-jΔt)] (5)

[0029] R(-jΔt)=R(jΔt) (6)

[0031] We obtain two equations (or systems of equations), namely:

[0032]

[0033]

[0034] The regular equation for the AR model is:

[0035]

[0036] In the formula, φ=[I,φ1,…,φ p ] T Let I be a ((p+1)×M) row M column matrix, where I is an M-order identity matrix; R N Same meaning as before; O pR is a (p×M) row, M column matrix with all elements being 0; R is a (p+1)×M row, (p+1)M column autocorrelation Toeplitz matrix, written in block matrix form, i.e.:

[0037]

[0038] Among them, R ij (mΔt) is an M×M square matrix, i = 1, ..., p+1; j = 1, ..., p+1; m = 0, ..., p, according to the Wiener-Khintchine formula:

[0039]

[0040] In the formula, f is the frequency of the fluctuating wind speed; S ij (f) is the autospectral density function of fluctuating wind speed at i = j; S ij (f) When i≠j, it is the cross-spectral density function of fluctuating wind speed, which can be derived from the auto-spectral density function S of fluctuating wind speed. ii (f) and coherence function r ij (f) Determine i = 1, ..., M, j = 1, ..., M;

[0041] According to the wind speed time history assumption formula (3), the repetition spectrum is 0, then

[0042]

[0043] In the formula, Sii(f) or Sjj(f) can take the form of Davenport spectrum, Simiu spectrum, Harris spectrum, Kaimal spectrum, Hino spectrum, etc., such as the Simiu spectrum:

[0044]

[0045] In the formula, v * Shear velocity, S ij Here, n represents the power spectral density of fluctuating wind speed, and K represents the frequency; K is the wind spectral coefficient. To calculate the average wind speed at the height of the point, v 10 The average wind speed at 10m is given.

[0046] From the self-spectral density function S of fluctuating wind speed ii (f) and coherence function r ij (f) Determine S ij (f) After that, substituting into formulas (9) and (10), the AR model coefficient matrix φ can be solved respectively. k The covariance matrix R NFor ease of calculation, it is usually assumed that the wind speed before the initial moment is 0, that is, when t≤0, V(t)=0. Based on this, N(t) is solved according to formula (2) and formula (3), and then substituted into formula (1) to obtain the horizontal fluctuating wind speed time history V(t).

[0047] Step 1.2: Generate average wind speed using the average wind speed exponential law;

[0048] The average wind speed at a height of 20 meters is set to 10 m / s, and the average wind speed index is as shown in formula (13).

[0049]

[0050] Where U(z) is the reference height z above the ground. r The average wind speed at the location; α is the ground roughness index, the value of which is related to the ground roughness; U(z) G (z) represents the gradient wind height. G The average wind speed at the location. According to the standard "Load Code for Design of Building Structures" (GB50009-2012), α = 0.3, U(z) = 0.3. G =550.

[0051] Step 1.3: Add the fluctuating wind speed to the average wind speed to obtain the total wind speed. According to Bernoulli's equation, convert the total wind speed into wind pressure and convert it into a concentrated force at the mass point, as shown in formula (14):

[0052]

[0053] Among them, w p Let v be the wind pressure at each point, and v be the wind speed at each point.

[0054] Step 1.4: Calculate the structural response under the applied equivalent concentrated force using the average acceleration method in the Newmark-β method. The calculation time step is 0.1s. The structural response includes displacement, velocity, and acceleration.

[0055] Step 1.5: Using the concentrated force as input and the structural response of each mass point as output, generate training and test samples. Create 10 training samples with a computation time of 80s each for training the LSTM. Create 10 test samples with computation times of 80s, 120s, 160s, 200s, 240s, 280s, 320s, 360s, 400s, and 440s for testing the generalization ability of the LSTM.

[0056] Preferably, the Simiu spectrum is used as the target spectrum to simulate the pulsating wind at a specified height. The pulsating wind speed is calculated every 1m of height, the calculation time for each height is 3000s, and the sampling frequency is 10Hz.

[0057] Preferably, the specific steps for constructing and training the LSTM in step 2 include:

[0058] Step 2.1 Construct an LSTM layer with 120 LSTM units, out_dim = 3, and use the tanh function as the activation function, where out_dim is the output dimension;

[0059] Step 2.2 uses the training samples from Step 1.5 to train the LSTM. Input and label values ​​are continuously extracted from the training samples and input into the LSTM to initialize it. The LSTM outputs data, and the optimizer continuously adjusts the learning rate. Finally, the model with the minimum loss on the training samples is used as the trained LSTM.

[0060] Preferably, the input value is the load, the label values ​​include displacement, velocity, and acceleration, and the output data includes displacement, velocity, and acceleration.

[0061] Preferably, Adam is used as the optimizer, with β1 = 0.9, β2 = 0.999, and learning rate a = 0.0001.

[0062] m Δs =β1m Δs-1 +(1-β1)g Δs (15)

[0064]

[0065]

[0066]

[0067]

[0068] Where Δs is the time step, and g Δs It is the gradient vector, m Δs and v Δs These are the first (mean) and second (non-central variance) bias moment estimates of the gradient, respectively. and Let β1 and β2 be the bias-corrected moment estimates for the first (mean) and second (non-central variance) gradients, respectively, where β1 and β2 are the exponential decay rates of the moment estimates, a is the learning rate, w is the model weights, and ε = 10e -8 .

[0069] Preferably, the physical information is obtained using structural dynamic equations, and the loss function of ordinary LSTM is used instead. P-LSTM As shown in Equation (20), and the Loss term of the dynamic equation for the predicted value as shown in Equation (21) SDEAdding them together, we obtain the LSTM loss function Loss for the fused structural dynamic equations. LSTM As shown in formula (22):

[0070]

[0071]

[0072] Loss LsTM =Loss P-LSTM +Loss SDE (twenty two)

[0074] Where n is the frequency, For the d-th prediction value of LSTM, The d-th true value of the LSTM; y pred P and M represent the acceleration, velocity, displacement, and load of each mass point in the LSTM-predicted structure, respectively; ass K and C represent the structural mass, stiffness, and damping, respectively.

[0075] The present invention has the following beneficial effects:

[0076] 1. This invention considers the structural dynamic equation residuals in the LSTM loss function, enabling the neural network to learn patterns from a large amount of data that conform to the structural dynamic equation, thus making the neural network more reliable.

[0077] 2. By predicting the structural displacement, velocity, and acceleration response of each mass point, the problem of existing neural networks only performing single-dimensional, i.e., single-mass point displacement prediction is solved, and multi-dimensional prediction needs can be met.

[0078] 3. By predicting the structural displacement, velocity, and acceleration response of each mass point, the residual terms of the structural dynamic equation in the loss function are directly substituted, avoiding the limitation of low-precision numerical differentiation required for single-mass point displacement prediction. Simultaneously, it enables high-precision prediction of all responses for all mass points. Attached Figure Description

[0079] Figure 1 A flowchart of an LSTM method for predicting structural response by incorporating physical information;

[0080] Figure 2 Diagram of the LSTM model;

[0081] Figure 3 This is a diagram of an LSTM cell model. Detailed Implementation

[0082] The specific embodiments of the present invention will be further described in detail below with reference to implementation examples. These implementation examples are only used to illustrate the present invention and do not limit the scope of the present invention. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the present invention; therefore, all equivalent technical solutions also fall within the scope of the present invention.

[0083] In this embodiment, please refer to Figure 1 A method for predicting structural response using LSTM that integrates physical information, comprising the following steps:

[0084] In one embodiment, training and test samples are generated based on the AR model and the Newmark-β method;

[0085] Step 1.1: Generate fluctuating wind speed based on the AR model;

[0086] Step 1.2: Generate average wind speed using the average wind speed exponential law;

[0087] Step 1.3: Add the fluctuating wind speed to the average wind speed to obtain the total wind speed. According to Bernoulli's equation, convert the total wind speed into wind pressure and convert it into a concentrated force at the mass point, as shown in formula (23):

[0088]

[0089] Among them, w p Let v be the wind pressure at each point, and v be the wind speed at each point.

[0090] Step 1.4: Calculate the structural response under the applied equivalent concentrated force using the average acceleration method in the Newmark-β method. The calculation time step is 0.1s. The structural response includes displacement, velocity, and acceleration.

[0091] Step 1.5: Using the concentrated force as input and the structural response of each mass point as output, generate training samples and test samples;

[0092] The Simiu spectrum was used as the target spectrum to simulate pulsating wind at a specified height. The pulsating wind speed was calculated every 1m of height, with a calculation time of 3000s for each height and a sampling frequency of 10Hz.

[0093] In one embodiment, an LSTM is constructed and trained, and the number of iterations is set to 1000, thereby causing the LSTM loss function to converge.

[0094] Please refer to Figure 2 Construct an LSTM layer with 120 LSTM units, out_dim = 3, and use the tanh activation function, where out_dim is the output dimension. Figure 3 As shown;

[0095] Step 2.2 uses the training samples from Step 1.5 to train the LSTM. Input and label values ​​are continuously extracted from the training samples and input into the LSTM to initialize it. The LSTM outputs data, and the optimizer continuously adjusts the learning rate. Finally, the model with the smallest loss on the training samples is used as the trained LSTM.

[0096] The input value is the load, and the label values ​​include displacement, velocity, and acceleration. The output data includes displacement, velocity, and acceleration.

[0097] In one embodiment, Adam is used as the optimizer, with β1 = 0.9, β2 = 0.999, and learning rate a = 0.0001. β1 and β2 are the exponential decay rates of the moment estimates.

[0098] In one embodiment, the loss function of a standard LSTM is... P-LSTM As shown in Equation (24), and the Loss term of the dynamic equation for the predicted value as shown in Equation (25) SDE Adding them together, we obtain the LSTM loss function Loss for the fused structural dynamic equations. LSTM As shown in formula (26):

[0099]

[0100]

[0101] Loss LSTM =Loss P-LSTM +Loss SDE (26)

[0103] Where n is the frequency, For the d-th prediction value of LSTM, The d-th true value of the LSTM; y pred P and M represent the acceleration, velocity, displacement, and load of each mass point in the LSTM-predicted structure, respectively; ass K and C represent the structural mass, stiffness, and damping, respectively.

[0104] In one embodiment, test samples are used to test the prediction accuracy and generalization ability of the trained LSTM, ensuring that the trained LSTM has sufficient prediction accuracy and the same accuracy for all samples.

[0105] Step 3.1 Predicts the test samples. The predicted single-degree-of-freedom structure fit is 99.99%, and the absolute error, mean absolute error, and mean absolute percentage error are all very small.

[0106] Step 3.2 Use the test sample with a duration of 80s in the test sample, input the load into LSTM, obtain the predicted displacement, velocity and acceleration, compare with the actual response value, evaluate the model prediction error, and test the remaining 10 training samples to verify that LSTM has good generalization ability.

[0107] In one embodiment, the load is input into the trained LSTM, and the LSTM outputs the predicted displacement, velocity, and acceleration.

[0108] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for predicting structural response using LSTM that integrates physical information, characterized by the following steps: include: Step 1: Generate training and test samples based on the AR model and the Newmark-β method; Step 2: Build and train the LSTM, and set the number of iterations to 1000 to bring the LSTM loss function to converge. Step 3: Use test samples to test the prediction accuracy and generalization ability of the trained LSTM to ensure that the trained LSTM has sufficient prediction accuracy and the same accuracy for all samples. Step 4: Input the load into the trained LSTM, and the LSTM will output the predicted displacement, velocity, and acceleration. The specific steps for generating training and test samples in step 1 include: Step 1.1: Generate fluctuating wind speed based on the AR model; Step 1.2: Generate average wind speed using the average wind speed exponential law; Step 1.3: Add the fluctuating wind speed to the average wind speed to obtain the total wind speed. According to Bernoulli's equation, convert the total wind speed into wind pressure and convert it into a concentrated force at the mass point, as shown in formula (1): （1） in, Let v be the wind pressure at each point, and v be the wind speed at each point. Step 1.4: Calculate the structural response under the applied equivalent concentrated force using the average acceleration method in the Newmark-β method. The calculation time step is 0.1s. The structural response includes displacement, velocity, and acceleration. Step 1.5: Using the concentrated force as input and the structural response of each mass point as output, generate training samples and test samples; The Simiu spectrum was used as the target spectrum to simulate pulsating wind at a specified height. The pulsating wind speed was calculated every 1m of height, with a calculation time of 3000s for each height and a sampling frequency of 10Hz. The specific steps for building and training the LSTM in step 2 include: Step 2.1 Construct an LSTM layer with 120 LSTM units, out_dim=3, and use the tanh function as the activation function, where out_dim is the output dimension; Step 2.2 uses the training samples from Step 1.5 to train the LSTM. Input and label values ​​are continuously extracted from the training samples and input into the LSTM to initialize it. The LSTM outputs data, and the optimizer continuously adjusts the learning rate. Finally, the model with the smallest loss on the training samples is used as the trained LSTM. The input value is the load, and the label values ​​include displacement, velocity, and acceleration. The output data includes displacement, velocity, and acceleration. Using Adam as the optimizer, set =0.9, =0.999, learning rate a=0.0001, and It is the exponential decay rate estimated by moments; Physical information includes structural dynamic equations, and the loss function of ordinary LSTM is used. P-LSTM As shown in Equation (2), and Equation (3), the residual term Loss of the dynamic equation for the predicted value. SDE Adding them together, we obtain the LSTM loss function Loss for the fused structural dynamic equations. LSTM As shown in formula (4): （2） （3） （4） Where n is the frequency, For the d-th prediction value of LSTM, The d-th true value of the LSTM; , , , The values ​​are the acceleration, velocity, displacement, and load of each mass point in the structure predicted by LSTM. K and C represent the structural mass, stiffness, and damping, respectively.

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