Structural seismic responses prediction using the gradient-enhanced hybrid PINN

Abstract

To rapidly and effectively assess the bridge seismic-resistant capability, it is essential to conduct efficient predictions of bridge seismic responses. Recently, physics informed neural network (PINN) has made great progress and utilized to solve differential equations in different fields. However, how to increase its accuracy and efficiency still remains an open challenge. In this work, a novel gradient-enhanced Fourth-Order Runge-Kutta PINN (gRK4-PINN), as a powerful hybrid PINN, is utilized to achieve this goal. As for gRK4-PINN, the physical information is not simply embedded into the loss function; instead, the RK4 method and the physical model is intricately integrated with the neural network. In addition, to improve the predictive performance, additional gradient equation is directly embedded in loss function. A large-span continuous girder high speed railway (CGHSR) bridge is adopted as numerical experiment to validate the fidelity of the proposed method. Results reveal that the Mean Absolute Error (MAE) of the predicting seismic responses is relatively small, whose value is below 0.014 in most of the time. These small MAE values indicate that the proposed gRK4-PINN performs well in predicting the seismic responses of the CGHSR bridge.

Keywords

deep learning high-speed railway bridge hybrid physics-informed neural network seismic response prediction Runge-Kutta method

Introduction

As the lifeline project, ensuring the structural safety of bridges is of paramount significance to the socio-economic development (Zhang and Alam, 2019). In recent years, the world has entered an active seismic period, with a series of devastating earthquakes occurring all over the world, including the Wenchuan Earthquake (Wang, 2008), the East Japan Earthquake (Krausmann and Cruz, 2013), and earthquakes in Turkey (Taftsoglou et al., 2023). The social and economic losses resulting from bridge-related seismic disasters have been significant (Zhong et al., 2023). How to rapidly and effectively assess the bridge seismic-resistant capability has become a key issue influencing the safety of transportation networks.

Acquiring the seismic response of bridges is pivotal for conducting comprehensive seismic analysis, such as vulnerability analysis (Rahman and Ullah, 2012), reliability analysis (Afshari et al., 2022), correction of nonlinear model (Touzé et al., 2021), et al. Traditionally, bridge seismic analysis relies on elastic-plastic finite element methods, which is time-consuming and exhibits poor real-time performance (Aghagholizadeh, 2020; Gönen and Soyöz, 2021; Krishnamoorthy and Anita, 2016). To mitigate the computational costs associated with bridge seismic analysis, surrogate models such as multiple regression models (Huang and Huang, 2020), Kriging models (Fuhg et al., 2021), ARMA models (Zhao et al., 2023), and support vector machines (Alizadeh et al., 2020) have been widely adopted. Due to the limited number of model parameters, these models are typically suitable for predicting the dynamic responses of simple linear systems, making them challenging to apply to complex dynamic systems (Barkhordari and Tehranizadeh, 2021; Oh et al., 2020; Wang et al., 2023). To this end, deep learning technology has been employed for predicting bridge seismic responses due to its excellent nonlinear fitting capabilities (Li et al., 2022; Liao et al., 2023). For instance, Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), and their improved variants have been utilized for seismic response prediction of engineering structures (Ning et al., 2023; Shokri and Tavakoli, 2019; Zhang R et al., 2019). Although these models have proven to be effective in predicting bridge seismic responses, two prominent shortcomings remain (Cai et al., 2021; Raissi et al., 2019): (a) the models heavily depend on extensive training data for accurate predictions; (b) neural networks lack clear and explicit physical interpretations, especially for deep network models.

In recent years, Physics-Informed Neural Networks (PINNs) have found extensive applications in solving differential equations (Blechschmidt and Ernst, 2021; Cuomo et al., 2022; Schiassi et al., 2021). By embedding physical information into the neural network, PINN exhibits superior predictive performance with limited training samples. It has been applied extensively in science and engineering, attaining significant success (Huang and Wang, 2022; Xu et al., 2023). Essentially, there exists two primary types of PINN, that is, the classic PINNs and the hybrid PINNs (Nascimento et al., 2020). As for the classic PINNs, the physical equations are added to the loss function as soft penalty constraints (Pan and Duraisamy, 2020). Alternatively, hybrid PINNs do not simply incorporate the physical information into the loss function; instead, the physical model is intricately integrated with the neural network (Yucesan and Viana, 2022). As a typical hybrid PINN, Fourth-Order Runge-Kutta PINN (RK4-PINN) is commonly employed for solving forward and inverse problems involving ordinary differential equations (Meng et al., 2020; Raissi et al., 2019). The unique structure of RK4-PINN neurons makes it outperform conventional CNN and RNN models. The physical model and neural network are deeply integrated in RK4-PINN, making the black-box network interpretable, which is a promising area in the future (Mattey and Ghosh, 2022; Zhong et al., 2022). Nevertheless, in aforementioned works, RK4-PINN was solely utilized to solve simple dynamic systems, whose potential application in seismic analysis for complex structures, such as high-rise buildings, large-scale bridges, et al., remains to be explored. Additionally, how to improve the efficiency and accuracy still remains an open challenge.

This work aims to establish an effective scheme to predict the seismic responses at critical locations of bridges. A large-span continuous girder high speed railway (CGHSR) bridge is adopted as the engineering background (Liang et al., 2021; Xing et al., 2022). The seismic responses at bridge critical locations are equivalent to that obtained by the equivalent system, whose unknown parameters are identified using the gradient-enhanced RK4-PINN (gRK4-PINN). The effectiveness of the method was validated through a numerical experiment. The remaining sections are arranged as follows. The following Section introduces the basic conception, including the governing equations and the architecture of the gRK4-PINN. Then, a numerical experiment is performed to test the fidelity of the method. Finally, conclusions are summarized.

Basic conception

Governing equations

The dynamic system under seismic excitation is utilized to illustrate the concept, which can be expressed as (Liang et al., 2021; Xing et al., 2022):

M \frac{d^{2} Y}{d t^{2}} (t) + C \frac{d Y}{d t} (t) + KY (t) = - M Γ \frac{d^{2} u_{g}}{d t^{2}} (t)

(1)

where M, C and K represents the mass, damping and stiffness matrices, respectively; Y,

\frac{d Y}{d t}

and

\frac{d^{2} Y}{d t^{2}}

are the relative displacement, velocity and acceleration, respectively;

\frac{d^{2} u_{g}}{d t^{2}}

denotes the seismic motion; Γ distributes the force.

The finite element models (FEMs) of the real complex engineering structures usually have large number of degrees of freedom (DoFs), that is, the mass, stiffness and damping matrices have extremely large scales, making the solution of Eq. (1) particularly challenging. Since the structural seismic response is the typical forced vibration, while the filtering effect of the site will result in the presence of predominant period in seismic motion. Thus, inspired by the Equivalent Linearization Method, the following Equivalent Single DoF (ESDoF) oscillator is utilized to model the seismic response at critical locations of the structure (Yazdanpanah et al., 2022).

m_{e q} \frac{d^{2} y}{d t^{2}} (t) + c_{e q} \frac{d y}{d t} (t) + k_{e q} y (t) = - m_{e q} \frac{d^{2} u_{g}}{d t^{2}} (t)

(2)

where m_eq, c_eq and k_eq represent the equivalent mass, damping and stiffness, respectively; y,

\frac{d y}{d t}

and

\frac{d^{2} y}{d t^{2}}

denote the displacement, velocity and acceleration at the concerned location under the seismic motion

\frac{d^{2} u_{g}}{d t^{2}}

, respectively. Once the system parameters, that is, m_eq, c_eq and k_eq, are determined, the structural seismic analysis can be easily executed.

Architecture of gRK4-PINN

To model and identify the ESDoF system in Eq. (2), the gRK4-PINN is adopted. Firstly, the Fourth-Order Runge-Kutta (RK4) method is briefly introduced. Consider the second order differential equation for the ESDoF system, that is, Eq. (2), which can be rewritten as (Press, 2007):

\frac{d^{2} y}{d t^{2}} (t) = \frac{1}{m_{e q}} [- m_{e q} \frac{d^{2} u_{g}}{d t^{2}} (t) - k_{e q} y (t) - c_{e q} \frac{d y}{d t} (t)] = F (y, \frac{d y}{d t}, t)

(3)

By letting

x = \frac{d y}{d t}

, Eq. (3) will reduce to the following equation:

\frac{d x}{d t} (t) = F (x, y, t)

(4)

Then the following recurrence formula is utilized to solve the concerned variables:

y_{i + 1} = y_{i} + \frac{h}{6} (X_{1} + 2 X_{2} + 2 X_{3} + X_{4})

(5)

x_{i + 1} = x_{i} + \frac{h}{6} (F_{1} + 2 F_{2} + 2 F_{3} + F_{4})

(6)

where h denotes the time step. The quantities in Eqs. (5) and (6) can be clearly clarified using Table 1. gRK4-PINN combines the RK4 method and the physics informed neural network (PINN) (Nascimento et al., 2020; Zhai et al., 2023). The architecture of gRK4-PINN is shown in Figure 1.

Table 1.

Quantities used in RK4 method.

t	y	$x = \frac{d y}{d t}$	$F = \frac{d x}{d t} = \frac{d^{2} y}{d t^{2}}$
T₁ = t_i	Y₁ = y_i	X₁ = x_i	F₁ = F(X₁,Y₁,T₁)
T₂ = t_i+h/2	$Y_{2} = y_{i} + \frac{h}{2} Y_{1}$	$X_{2} = x_{i} + \frac{h}{2} F_{1}$	F₂ = F(X₂,Y₂,T₂)
T₃ = t_i+h/2	$Y_{3} = y_{i} + \frac{h}{2} Y_{2}$	$X_{3} = x_{i} + \frac{h}{2} F_{2}$	F₃ = F(X₃,Y₃,T₃)
T₄ = t_i+h	Y₄ = y_i+hY₃	X₄ = x_i+hF₃	F₄ = F(X₄,Y₄,T₄)

Figure 1.

Architecture of gRK4-PINN.

From Eqs. (5) and (6), X are recursively fed back into F four times at each time step. Thus, function F is the key information in RK4 method. Here, since the system parameters remain to be determined, function F is only partially known. To this end, the gRK4-PINN is utilized to identify the ESDoF system.

Table 1 presents the solution process of the RK4 method, with the parameters explained analogous to Eqs. (2) to (6). As shown in Figure 1, the general structure of gRK4-PINN is similar to Recurrent Neural Network, in which the ground motion $\frac{d^{2} u_{g}}{d t^{2}}$ are input data, while the concerned seismic displacement responses are set as outputs. The ground truth $\frac{d^{2} y}{d t^{2}}$ is utilized to constrain the predicting F, while the predicting F is reversely utilized in the RK4 cell to update the system parameters and other quantities utilized in Eq. (3), that is, the physical information. Using Π denotes the mapping function of the gRK4-PINN, then the prediction loss can be calculated as:

L_{t o t a l} = α_{1} L_{d a t a} + α_{2} L_{p h y s i c s} + α_{3} L_{g_p h y s i c s}

(7)

L_{d a t a} = \frac{1}{N} {\sum_{i = 1}^{N} (\frac{d^{2} y}{d t^{2}} (t_{i}) - Π (\frac{d^{2} u_{g}}{d t^{2}} (t_{i}); ϑ))}^{2}

(8)

L_{p h y s i c s} = \frac{1}{N} {\sum_{i = 1}^{N} (Π (\frac{d^{2} u_{g}}{d t^{2}} (t_{i}); ϑ) - \frac{1}{m_{e q}} [- m_{e q} \frac{d^{2} u_{g}}{d t^{2}} (t_{i}) - k_{e q} y (t_{i}) - c_{e q} \frac{d y}{d t} (t_{i})])}^{2}

(9)

L_{g_p h y s i c s} = \frac{d L_{p h y s i c s}}{d t}

(10)

where ϑ is the network parameters; α_i,i=1,2,3 represent the weight coefficients; N denotes the total calculating time steps. RMSprop optimizer in TensorFlow is utilized to conduct the training process. Different from the commonly used RK4-PINN, the additional gradient loss, that is,

L_{g_p h y s i c s}

, are included in the total loss. It is noted that, unlike seismic solutions based on finite element analysis, deep learning methods only require input of actual seismic motion data. There is no need to consider the multi-point input issues present in finite element models. The prediction accuracy of seismic response depends entirely on the optimization of the physical and data loss functions.

Numerical experiment

To demonstrate the effectiveness and give more details when utilizing the gRK4-PINN, numerical experiments of seismic responses prediction for a large-span continuous girder high speed railway (CGHSR) bridge are designed.

Engineering background

In this section, a CGHSR bridge under seismic excitation is selected to elucidate the effectiveness of the gRK4-PINN. Figure 2 shows the structure layout of the CGHSR bridge, consists of a cast-in-place, non-uniform cross-section concrete box girder with a layout of 68m + 132m + 68m. The bridge girder is constructed using C55 concrete. Four solid reinforced shaft piers are constructed using C35 concrete, which are numbered from pier 1# to pier 4#.

Figure 2.

Layout of the CGHSR bridge configuration (Unit: m).

Then, a lumped mass FEM is established using commercial code ANSYS, as shown in Figure 3. Some fundamental model parameters are summarized in Table 2.

Figure 3.

FEM of the CGHSR bridge.

Table 2.

Fundamental parameters of the FEM.

Definition	Value
Total weight	214800 kN
Young’s modulus of girder concrete	3.55×10⁴ MPa
Young’s modulus of pier concrete	3.15×10⁴ MPa
Poisson’s ratio of concrete	0.2
Element for girder and piers	BEAM4
Element for bearing connects pier and girder	COMBIN14
Number of nodes	67
Number of DoFs	374

In Figure 3, the DoFs of each bottom of pier is fixed, while the longitudinal DoF(x) of piers and the main girder in their connection parts are released except for pier 2#. The modal analysis is performed and compared with the design value, as listed in Table 3, indicating the reliability of the FEM. More details of the CGHSR bridge can be found in authors’ previous works (Liang et al., 2021; Xing et al., 2022).

Table 3.

Summary of the first five modes of the CGHSR bridge.

No.	Frequency (Hz)		Characteristics of modes
No.	FEM	Design	Characteristics of modes
1	1.1291	1.0096	Symmetric lateral bending vibration of the girder
2	1.1774	1.1334	Longitudinal bending vibration of the pier
3	1.1824	1.2705	Symmetric vertical bending vibration of the girder
4	1.9482	1.5872	Lateral bending vibration of one side span
5	2.1495	1.7277	Lateral bending vibration of the other side span

Generation of the ground motions

Consider the following two-sided seismic evolutionary power spectral density (EPSD)

S_{u_{g}} (t, ω)

(Cacciola and Deodatis, 2011):

\begin{array}{l} S_{u_{g}} (t, ω) = {{[\frac{t}{c} e^{(1 - \frac{t}{c})}]}^{d}}^{2} \cdot \frac{ω_{g}^{4} (t) + 4 ξ_{g}^{2} (t) ω_{g}^{2} (t) ω^{2}}{{[ω^{2} - ω_{g}^{2} (t)]}^{2} + 4 ξ_{g}^{2} (t) ω_{g}^{2} (t) ω^{2}} \cdot \\ \frac{ω^{4}}{{[ω^{2} - ω_{f}^{2} (t)]}^{2} + 4 ξ_{f}^{2} (t) ω_{f}^{2} (t) ω^{2}} \cdot \frac{{\bar{a}}_{\max}^{2}}{γ^{2} [π ω_{g} (t) (2 ξ_{g} (t) + \frac{1}{2 ξ_{g} (t)})]} \end{array}

(11)

\begin{array}{l} ω_{g} (t) = ω_{0} - a \frac{t}{T}, ξ_{g} (t) = ξ_{0} + b \frac{t}{T} \\ ω_{f} (t) = 0.1 ω_{g} (t), ξ_{f} (t) = ξ_{g} (t) \end{array}

(12)

where c is the occurrence time of the peak ground excitation; d represents the shape factor; T is the total time duration of the ground motion. The dimension-reduced spectral representation method is utilized to generate 144 samples of the ground motion (Liu et al., 2015). Some fundamental parameters for the simulation are summarized in Table 4.

Table 4.

Fundamental parameters for generating ground motion.

Definition	Value	Definition	Value
ω ₀	13.5 rad	d	2
ξ ₀	0.65	T	25
a	5	${\bar{a}}_{\max}^{2}$	196 cm/s²
b	0.2	γ	2.6
c	6	Cut-off frequency	219.9 rad/s
Number of frequency	1800	Frequency interval	0.12 rad/s

The generated representative samples are shown in Figure 4. What’s more, the standard deviation and the ensemble response spectrum of the generated samples are compared with the targets, as shown in Figure 5.

Figure 4.

Generated representative samples.

Figure 5.

Comparison between the standard deviation and the ensemble response spectrum of the generated samples with the target.

As shown in Figures 4 and 5, the generated ground motion has reasonable amplitude, while the ensemble standard deviation and the response spectrum agree well with targets. This demonstrates that the generated samples can be further utilized for structural seismic responses analysis.

Seismic response prediction

Using the generated ground motions in above section, the seismic response analysis of the bridge can be performed. The seismic response at the top of the pier 2# is selected as the interesting quantity, which is utilized as targets to validate the validity of the predicting response obtained by gRK4-PINN. The parameters of the ESDoF system in Eq. (3) are identified using 20 randomly selected generated ground motions and the corresponding seismic responses. The mean of the identified m_eq, c_eq and k_eq are 5.823×10⁴ kg, 5.572×10⁴ kg/s and 5.537×10⁶ kg/s², respectively. The remaining samples were used as the test set to validate the performance of the model. It is noted that predicting the displacement responses of the remaining 124 samples takes no more than 10 s, while using finite element methods requires several hours. This indicates that employing deep learning methods offers significantly higher computational efficiency. Several representative predicting samples are randomly selected and compared with the targets, that is, the numerical results obtained by the seismic analysis of the bridge FEM, as shown in Figure 6. Furthermore, using the same training samples, the seismic response was predicted by the RK4-PINN network, results are also exhibited in Figure 6. Note that the displacement data fed in the neural network are contaminated with Gaussian noise with zero mean and 2×10^-3 standard deviation.

Figure 6.

gRK4-PINN predicting results with noisy training samples.

It can be seen from Figure 6 that the predictions converge to the noisy training samples, filtering the noise in training data, indicating the gRK4-PINN can effectively identify the ESDoF system parameters and achieve relatively accurate predictions. However, the RK4-PINN prediction results exhibit a certain deviation from the targets, suggesting that, when using the same training samples, the gradient loss function is advantageous for enhancing the training accuracy. What’s more, the Mean Absolute Error (MAE), which is a common metric to evaluate the network performance, is also exhibited. It is evident that the MAE of the gRK4-PINN results is relatively smaller than that of RK4-PINN. For example, the MAE is 0.0027 for gRK4-PINN in sample 1, which is 0.0074 for RK4-PINN. This further confirms the importance of the gradient loss function in network training. To further elucidate the performance of the hybrid PINN, the frequency distribution of the MAE for all samples is statistically analyzed, as shown in Figure 7.

Figure 7.

Frequency distribution of MAE for all samples.

As shown in Figure 7, the MAE is below 0.014 in most of the time. These values are sufficiently small, indicating the well performance of the hybrid PINN in predicting bridge seismic responses.

Conclusions

In this work, a novel gradient-enhanced RK4-PINN is developed to conduct the seismic prediction of the CGHSR bridge. The bridge seismic response at critical locations is computed through Eq. (3) emanated from the concept of a SDoF system. The system unknown parameters are identified using the gRK4-PINN. The effectiveness and efficiency of the proposed methods are demonstrated via a numerical experiment. Conclusions can be drawn as follows.

(1) gRK4-PINN can effectively identify the unknown parameters of the ESDoF system. In this case, the mean of the identified m_eq, c_eq and k_eq are 5.823×10⁴ kg, 5.572×10⁴ kg/s and 5.537×10⁶ kg/s², respectively.

(2) gRK4-PINN demonstrates a certain level of robustness. The predicting results converge to the noisy training samples, filtering the noise in training data.

(3) The gRK4-PINN can achieve relatively accurate predictions. The MAE of the predictions is below 0.008, while never above 0.014 in most of the time.

The gRK4-PINN network model still faces several challenges that limit its further application in the field of structural seismic prediction: a) The selection of model parameters can significantly impact prediction accuracy, often requiring empirical knowledge or extensive experimentation to determine optimal values; b) When dealing with complex structural systems, various seismic loading scenarios, or nonlinear effects, the generalization ability of the model may decrease, leading to diminished predictive performance; c) Training processes for complex structural models and large-scale datasets typically demand substantial computational resources.

Footnotes

Author contributions

Conceptualization, Funding acquisition, Writing - original draft. Software, Funding acquisition, Formal analysis. Supervision, Funding acquisition, Data curation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research work was jointly supported by the National Natural Science Foundation of China (Nos. 52208468, 52208481 and 52338011), the China Postdoctoral Science Foundation (No. 2023M730581), the Natural Science Foundation of Jiangsu Province (No. BK20220851) and the Fundamental Research Funds for the Central Universities.

ORCID iD

Hao Wang

References

Afshari

Enayatollahi

, et al. (2022) Machine learning-based methods in structural reliability analysis: a review. Reliability Engineering and System Safety 219: 108223.

Aghagholizadeh

(2020) A finite element model for seismic response analysis of vertically-damped rocking-columns. Engineering Structures 219: 110894.

Alizadeh

Allen

Mistree

(2020) Managing computational complexity using surrogate models: a critical review. Research in Engineering Design 31: 275–298.

Barkhordari

Tehranizadeh

(2021) Response estimation of reinforced concrete shear walls using artificial neural network and simulated annealing algorithm. Structures 34: 1155–1168.

Blechschmidt

Ernst

(2021) Three ways to solve partial differential equations with neural networks-A review. GAMM-Mitteilungen 44(2): e202100006.

Cacciola

Deodatis

(2011) A method for generating fully non-stationary and spectrum-compatible ground motion vector processes. Soil Dynamics and Earthquake Engineering 31(3): 351–360.

Cai

Mao

Wang

, et al. (2021) Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta Mechanica Sinica 37(12): 1727–1738.

Cuomo

Di Cola

Giampaolo

, et al. (2022) Scientific machine learning through physics-informed neural networks: where we are and what’s next. Journal of Scientific Computing 92(3): 88.

Fuhg

Fau

Nackenhorst

(2021) State-of-the-art and comparative review of adaptive sampling methods for kriging. Archives of Computational Methods in Engineering 28: 2689–2747.

10.

Gönen

Soyöz

(2021) Seismic analysis of a masonry arch bridge using multiple methodologies. Engineering Structures 226: 111354.

11.

Huang

(2020) Predicting capacity model and seismic fragility estimation for RC bridge based on artificial neural network. Structures 27: 1930–1939.

12.

Huang

Wang

(2022) Applications of physics-informed neural networks in power systems-a review. IEEE Transactions on Power Systems 38(1): 572–588.

13.

Krausmann

Cruz

(2013) Impact of the 11 March 2011, Great East Japan earthquake and tsunami on the chemical industry. Natural Hazards 67: 811–828.

14.

Krishnamoorthy

Anita

(2016) Soil-structure interaction analysis of a FPS-isolated structure using finite element model. Structures 5: 44–57.

15.

Chen

(2022) Fast seismic response estimation of tall pier bridges based on deep learning techniques. Engineering Structures 266: 114566.

16.

Liang

Wang

, et al. (2021) Multiple tuned inerter-based dampers for seismic response mitigation of continuous girder bridges. Soil Dynamics and Earthquake Engineering 151: 106954.

17.

Liao

Lin

Zhang

, et al. (2023) Attention-based LSTM (AttLSTM) neural network for seismic response modeling of bridges. Computers and Structures 275: 106915.

18.

Liu

Zeng

(2015) Simulation of non-stationary ground motion by spectral representation and random functions. Journal of Vibration Engineering 28(3): 411–417, (in Chinese).

19.

Mattey

Ghosh

(2022) A novel sequential method to train physics informed neural networks for allen Cahn and Cahn Hilliard equations. Computer Methods in Applied Mechanics and Engineering 390: 114474.

20.

Meng

Zhang

, et al. (2020) PPINN: Parareal physics-informed neural network for time-dependent PDEs. Computer Methods in Applied Mechanics and Engineering 370: 113250.

21.

Nascimento

Fricke

Viana

FAC

(2020) A tutorial on solving ordinary differential equations using Python and hybrid physics-informed neural network. Engineering Applications of Artificial Intelligence 96: 103996.

22.

Ning

Xie

Sun

(2023) LSTM, WaveNet, and 2D CNN for nonlinear time history prediction of seismic responses. Engineering Structures 286: 116083.

23.

Glisic

Park

, et al. (2020) Neural network-based seismic response prediction model for building structures using artificial earthquakes. Journal of Sound and Vibration 468: 115109.

24.

Pan

Duraisamy

(2020) Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability. SIAM Journal on Applied Dynamical Systems 19(1): 480–509.

25.

Press WH (2007) Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge: Cambridge University Press.

26.

Rahman

Ullah

(2012) Seismic vulnerability assessment of RC structures: a review. International Journal of Science and Emerging Technologies 4(4): 171–177.

27.

Raissi

Perdikaris

Karniadakis

(2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378: 686–707.

28.

Schiassi

Furfaro

Leake

, et al. (2021) Extreme theory of functional connections: a fast physics-informed neural network method for solving ordinary and partial differential equations. Neurocomputing 457: 334–356.

29.

Shokri

Tavakoli

(2019) A review on the artificial neural network approach to analysis and prediction of seismic damage in infrastructure. International Journal of Hydromechatronics 2(4): 178–196.

30.

Taftsoglou

Valkaniotis

Papathanassiou

, et al. (2023) Satellite imagery for rapid detection of liquefaction surface manifestations: the case study of Türkiye-Syria 2023 earthquakes. Remote Sensing 15(17): 4190.

31.

Touzé

Vizzaccaro

Thomas

(2021) Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105(2): 1141–1190.

32.

Wang

(2008) A preliminary report on the great Wenchuan earthquake. Earthquake Engineering and Engineering Vibration 7: 225–234.

33.

Wang

Noori

, et al. (2023) Seismic response prediction of structures based on Runge-Kutta recurrent neural network with prior knowledge. Engineering Structures 279: 115576.

34.

Xing

Wang

, et al. (2022) Stochastic analysis of a large-span continuous girder high-speed railway bridge under fully non-stationary earthquake. Applied Sciences 12(24): 12684.

35.

Wang

Xing

, et al. (2023) Physics guided wavelet convolutional neural network for wind-induced vibration modeling with application to structural dynamic reliability analysis. Engineering Structures 297: 117027.

36.

Yazdanpanah

Chang

Park

, et al. (2022) Seismic response prediction of RC bridge piers through stacked long short-term memory network. Structures 45: 1990–2006.

37.

Yucesan

Viana

FAC

(2022) A hybrid physics-informed neural network for main bearing fatigue prognosis under grease quality variation. Mechanical Systems and Signal Processing 171: 108875.

38.

Zhai

Tao

Bao

(2023) Parameter estimation and modeling of nonlinear dynamical systems based on Runge-Kutta physics-informed neural network. Nonlinear Dynamics 111: 21117–21130.

39.

Zhang

Alam

(2019) Performance-based seismic design of bridges: a global perspective and critical review of past, present and future directions. Structure and Infrastructure Engineering 15(4): 539–554.

40.

Zhang

Chen

, et al. (2019) Deep long short-term memory networks for nonlinear structural seismic response prediction. Computers and Structures 220: 55–68.

41.

Zhao

Jiang

Vega

, et al. (2023) Surrogate modeling of nonlinear dynamic systems: a comparative study. Journal of Computing and Information Science in Engineering 23(1): 011001.

42.

Zhong

Wang

(2022) Low-temperature plasma simulation based on physics-informed neural networks: frameworks and preliminary applications. Physics of Fluids 34(8): 087116.

43.

Zhong

Mao

Yuan

(2023) Lifetime seismic risk assessment of bridges with construction and aging considerations. Structures 47: 2259–2272.