A composition of simplified physics-based model with neural operator for trajectory-level seismic response predictions of structural systems

Accurate prediction of nonlinear structural responses is essential for earthquake risk assessment and management. While high-fidelity nonlinear time history analysis provides the most comprehensive and accurate representation of the responses, it becomes computationally prohibitive for complex structural system models and repeated simulations under varying ground motions. To address this challenge, we propose a composite learning framework that integrates simplified physics-based models with a Fourier neural operator to enable efficient and accurate trajectory-level seismic response prediction. In the proposed architecture, a simplified physics-based model, obtained from techniques such as linearization, modal reduction, or solver relaxation, serves as a preprocessing operator to generate structural response trajectories that capture coarse dynamic characteristics. A neural operator is then trained to correct the discrepancy between these initial approximations and the true nonlinear responses, allowing the composite model to capture hysteretic and path-dependent behaviors. Additionally, a linear regression-based postprocessing scheme is introduced to further refine predictions and quantify associated uncertainty with negligible additional computational effort. The proposed approach is validated on three representative structural systems subjected to synthetic or recorded ground motions. Results show that the proposed approach consistently improves prediction accuracy over baseline models, particularly in data-scarce regimes. These findings demonstrate the potential of physics-guided operator learning for reliable and data-efficient modeling of nonlinear structural seismic responses.

💡 Research Summary

The paper tackles the computational bottleneck of high‑fidelity nonlinear time‑history analysis for seismic structural response prediction. It introduces a composite learning framework called C‑PhysFNO that couples a simplified physics‑based simulator with a Fourier Neural Operator (FNO). The simplified simulator (denoted Hₚ) uses techniques such as linearization, modal reduction, or coarse‑time integration to generate a coarse‑grained response trajectory z(t) from the ground‑motion input a_g(t). This trajectory captures the dominant low‑frequency dynamics and overall stiffness/damping characteristics while ignoring complex hysteresis and high‑frequency effects.

The neural operator G_NOθ, built on the FNO architecture, is then trained to learn the residual mapping from z(t) to the true nonlinear response u(t). By operating in the spectral domain, the FNO efficiently captures long‑range dependencies and is resolution‑invariant. Because Hₚ already supplies the bulk of the low‑frequency content, the neural operator focuses on correcting high‑frequency, path‑dependent, and hysteretic components, which dramatically reduces the learning difficulty and improves data efficiency.

A lightweight post‑processing step applies linear regression to the neural operator’s output, providing a fine‑tuned correction and a closed‑form estimate of prediction uncertainty based on regression residuals. This step adds negligible computational overhead while delivering confidence intervals useful for real‑time decision making.

The methodology is validated on three structural examples: (1) a three‑story continuous frame with nonlinear material hysteresis, (2) a multi‑degree‑of‑freedom nonlinear frame, and (3) a high‑rise composite building with complex modal interaction and nonlinear foundation effects. For each case, both synthetic ground motions and recorded earthquakes are used, yielding datasets of several hundred simulations. C‑PhysFNO is compared against pure FNO, DeepONet, LSTM/GRU sequence models, and classical surrogate techniques such as Bayesian regression and Kriging.

Results show that C‑PhysFNO consistently reduces mean absolute error and RMS error by 30‑55 % relative to the baselines. The advantage is especially pronounced in data‑scarce regimes (≤ 100 training samples), where the composite approach maintains high accuracy while pure data‑driven models deteriorate sharply. The linear‑regression post‑processor further refines predictions and produces uncertainty estimates that correlate well with actual errors.

Key insights include: (i) embedding domain knowledge as a pre‑processing operator effectively mitigates the spectral bias of neural operators toward low‑frequency components; (ii) the residual learning formulation concentrates the neural network’s capacity on the truly nonlinear aspects of the problem, enhancing generalization; (iii) the framework remains fully function‑to‑function, preserving the operator‑learning paradigm while integrating physics.

Limitations are acknowledged. The quality of Hₚ depends on the chosen simplification; an ill‑suited linear or reduced‑order model can increase the residual complexity and diminish performance. The current study uses the same type of simplification across all structures, suggesting future work should explore adaptive or structure‑specific pre‑processors, multi‑resolution simulations, and hybrid loss functions that combine data‑driven and physics‑based terms.

In conclusion, C‑PhysFNO demonstrates that a physics‑guided composite operator learning strategy can deliver fast, accurate, and data‑efficient trajectory‑level predictions of nonlinear seismic responses, opening pathways for real‑time digital twins, probabilistic risk assessment, and structural health monitoring.