Neural Operators for Stochastic Modeling of Nonlinear Structural System Response to Natural Hazards

TL;DR

This work advances neural-operator metamodeling for nonlinear structural dynamics under stochastic natural hazards by introducing two novel architectures: DeepFNOnet, which hybrids DeepONet and FNO to overcome spectral bias and improve long-time predictions, and SA-FNO, which uses self-adaptive temporal weights to enhance high-frequency response accuracy. Through seismic and wind case studies, the authors demonstrate that these operators achieve high predictive accuracy while offering orders-of-magnitude speedups over high-fidelity simulations. The results indicate strong potential for real-time hazard forecasting and large-scale uncertainty propagation in civil engineering, enabling rapid risk assessment and resilience optimization. The study also provides a practical framework for deploying operator-learning surrogates in engineering workflows, with data and code to be released publicly.

Abstract

Traditionally, neural networks have been employed to learn the mapping between finite-dimensional Euclidean spaces. However, recent research has opened up new horizons, focusing on the utilization of deep neural networks to learn operators capable of mapping infinite-dimensional function spaces. In this work, we employ two state-of-the-art neural operators, the deep operator network (DeepONet) and the Fourier neural operator (FNO) for the prediction of the nonlinear time history response of structural systems exposed to natural hazards, such as earthquakes and wind. Specifically, we propose two architectures, a self-adaptive FNO and a Fast Fourier Transform-based DeepONet (DeepFNOnet), where we employ a FNO beyond the DeepONet to learn the discrepancy between the ground truth and the solution predicted by the DeepONet. To demonstrate the efficiency and applicability of the architectures, two problems are considered. In the first, we use the proposed model to predict the seismic nonlinear dynamic response of a six-story shear building subject to stochastic ground motions. In the second problem, we employ the operators to predict the wind-induced nonlinear dynamic response of a high-rise building while explicitly accounting for the stochastic nature of the wind excitation. In both cases, the trained metamodels achieve high accuracy while being orders of magnitude faster than their corresponding high-fidelity models.

Figures (10)

• Figure 1: DeepONet with an FNN architecture for both the branch and the trunk networks. The element-wise dot product of the output feature embedding of the branch net, $[b_1, b_2, \ldots, b_p]^\intercal \in \mathbb{R}^p$, and the trunk net, $[c_1, c_2, \ldots, c_p]^\intercal \in \mathbb{R}^p$, yields the solution operator, $\mathcal{G}_{\bm\theta}$, where $\bm{\theta}$ denotes the trainable parameters of the network. The loss function, $\mathcal{L}(\bm{\theta})$, which is obtained as the difference between the exact and predicted solution over all spatial and temporal points, is minimized to obtain the optimized parameters of the network, $\bm{\theta}^{*}$. For any point $\zeta=\{x_j, y_j, z_j, t_j\}$, the output $\mathcal{G}(u)(\zeta) \in \mathbb{R}$ is a real number.
• Figure 2: Schematic of the FNO li2020fourier. The FNO takes uniformly discretized temporal inputs and associated features, such as ground motions in seismic engineering problems or dynamic wind loads in wind engineering problems, at each floor level of the discretized building system. These inputs are first projected into a higher-dimensional feature space, $\mathcal{P}$, via a shallow neural network. The transformed representation is then processed through multiple Fourier layers, each comprising a forward Fast Fourier Transform (FFT), a linear transformation of low-frequency modes, and an inverse FFT. After each layer, the output is augmented with a learned weight matrix and passed through an activation function to introduce nonlinearity. Vectorized outputs are enabled through the use of multiple neurons in the shallow neural network, $\mathcal{Q}$.
• Figure 3: Schematic representation of DeepFNOnet, a hybrid neural architecture employing a two-stage training methodology. The framework first follows the conventional DeepONet training procedure, subsequently leveraging DeepONet's predictions by concatenating them with the original input time signals as enhanced inputs to the FNO. This integrated approach enables dual capabilities: temporal interpolation through the DeepONet framework and one-shot super-resolution facilitated by the FNO architecture, thereby synergistically combining the strengths of both neural operators.
• Figure 4: Schematic representation of the 6-story nonlinear shear building subjected to stochastic ground motions.
• Figure 5: Comparative illustration of the prediction accuracies of FNO and SA-FNO for each floor of the shear building example. The left-hand panel illustrates the building setup. The middle panel reports the comparison of the floor responses predicted by the FNO and SA-FNO to the ground truth. The right panel displays the error measure, $e$, plotted in logarithmic scale on the vertical axis. The bottom plot shows the applied base acceleration time history.