A Physics-Informed U-net-LSTM Network for Data-Driven Seismic Response Modeling of Structures

TL;DR

This work tackles the challenge of achieving accurate yet efficient seismic response predictions for nonlinear structures by blending physics with deep learning. The authors propose PhyULSTM, a hybrid architecture that couples a causal 1D U‑Net with a deep LSTM and a graph-based differentiator, guided by a physics-informed loss that enforces the equations of motion. Through numerical and experimental validation, PhyULSTM demonstrates superior accuracy and generalization, including scenarios with limited data or only acceleration measurements, outperforming the PhyCNN baseline. The approach offers a practical surrogate for real-time seismic analysis and structural health monitoring, capable of handling incomplete constitutive knowledge while maintaining physical consistency.

Abstract

Accurate and efficient seismic response prediction is essential for the design of resilient structures. While the Finite Element Method (FEM) remains the standard for nonlinear seismic analysis, its high computational demands limit its scalability and real time applicability. Recent developments in deep learning, particularly Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), and Long Short Term Memory (LSTM) models, have shown promise in reducing the computational cost of nonlinear seismic analysis of structures. However, these data driven models often struggle to generalize and capture the underlying physics, leading to reduced reliability. We propose a novel Physics Informed U Net LSTM framework that integrates physical laws with deep learning to enhance both accuracy and efficiency. By embedding domain specific constraints into the learning process, the proposed model achieves improved predictive performance over conventional Machine Learning architectures. This hybrid approach bridges the gap between purely data driven methods and physics based modeling, offering a robust and computationally efficient alternative for seismic response prediction of structures.

Figures (12)

• Figure 1: The proposed Physics-Informed U-net-LSTM Network (PhyULSTM) for time-series modeling. The 1D U-net first receives the ground acceleration data as input and extracts features at multiple time scales and adapts it to our auto-regressive scenario by ensuring that the convolutions are causal. The outputs from the U-net are then fed to the deep LSTM network that maps the temporal feature maps to the corresponding output space. The outputs are state space variables ${z}(t)$ = $\{x (t), \dot{x}(t), g(t)\}$. Available physics knowledge is incorporated directly into the loss function. In addition to it, a graph-based tensor differentiator using the central finite difference method developed by Zhang et.al in their PhyCNN framework is implemented to calculate the derivative of state space outputs ${z_t}(t)$ = $\{x_{t} (t), \dot{x}_{t}(t), g_t (t)\}$ to construct the physics loss from the governing equation, where the subscript t represents the derivative of the state with respect to time. By optimizing the network hyperparameters $\theta=\{W_{\theta},b_{\theta} \}$, PhyULSTM can interpret the measurement data (e.g. $\{x_m, \dot{x}_m, g_m\}$) While satisfying the physical equation of motion in equation \ref{['eq:gov']}, e.g. $f\rightarrow 0$. Here $W_{\theta}$,$b_{\theta}$ are weights and biases of the neural network. $J_D(\theta)$ is the data loss based on the measurements and $J_P(\theta)$ denotes the physics loss, which imposes a physical constraint on the neural network.
• Figure 2: Proposed 1D version of U-net: The U-Net architecture is mainly divided into four major components: the encoder blocks, bottleneck, decoder blocks, and the output convolution block. The encoder blocks perform convolution and pooling operations, generating skip connections and providing input to subsequent encoder blocks after pooling. The decoder blocks involve upsampling, followed by concatenation with the skip connections from the corresponding encoder block outputs, and subsequent convolution operations. The bottleneck layer, serving as a bridge, consists solely of a convolution block, linking the encoder and decoder paths. The output from the final decoder block is fed to the output convolution block, where it first passes through a Conv1D layer with a number of filters equal to the number of channels required in the final output (kernel size = 1), and sigmoid activation, followed by a second Conv1D layer and a linear activation function, which further processes the output to format the final output as a three-dimensional array, where the entries are sampled in the first dimension, time history steps in the second dimension, and output features in the last dimension. Here, ‘ ConvBlock’ is a fundamental building block consisting of a convolutional layer, followed by Batch Normalisation, and a ReLU activation function.
• Figure 3: Schematic of deep LSTM networks: (a) corresponds to the network architecture of a deep LSTM network featuring $m$ LSTM layers and multiple fully-connected layers for sequence-to-sequence modeling and (b) corresponds to the architecture of a typical LSTM cell at the $l$th layer and time $t$, showing cell input $X^{(l)}_t$, cell output $Y^{(l)}_t$, cell state $c^{(l)}_t$, hidden state $h^{(l)}_t$, and gate variables $\{f^{(l)}_t, i^{(l)}_t, \tilde{c}^{(l)}_t, o^{(l)}_t\}$, respectively.
• Figure 4: Time histories of displacement responses predicted by PhyULSTM (red, left column) and PhyCNN (red, right column) compared against the reference solution (blue). Each row corresponds to a different earthquake input (1, 2, and 3). Subplots are labeled 1a, 1b, 2a, 2b, 3a, and 3b for reference, where the left subplots (a) show PhyULSTM predictions and the right subplots (b) show PhyCNN predictions. Across all three test cases, PhyULSTM consistently tracks the true system behavior with higher accuracy, capturing both transient peaks and steady-state oscillations. In contrast, PhyCNN fails to reproduce finer-scale dynamics and underestimates displacement amplitudes during critical phases.
• Figure 5: Predicted time histories of velocity ($\dot{x}$) and nonlinear restoring force ($g$) for a representative ground motion are shown.The left column shows the responses predicted by PhyULSTM, and the right column shows those predicted by PhyCNN. Subplots are labeled 4a, 4b, 4c, and 4d for reference: 4a and 4c correspond to velocity and normalized restoring force predicted by PhyULSTM, while 4b and 4d correspond to the same predicted by PhyCNN. The PhyULSTM model captures both quantities with high fidelity compared to the numerical reference, successfully reproducing subtle fluctuations and transient peaks.