Fourier Neural Operators for Structural Dynamics Models: Challenges, Limitations and Advantages of Using a Spectrogram Loss

2025-11-11

AI Summary

Fourier Neural Operators (FNOs) excel in modeling linear systems but suffer from artificial energy dissipation, spectral distortion, and low-frequency bias when applied to nonlinear, nonstationary systems—e.g., structural dynamics—defects that persist regardless of training data scale. To address these limitations, we propose a spectral-graph-guided loss function that jointly optimizes time-domain mean squared error with frequency-domain amplitude and phase errors, coupled with a spectral-aware training strategy to suppress numerical dissipation and enhance energy conservation and phase fidelity. Evaluated on high-degree-of-freedom nonlinear systems—including the IEA 15-MW wind turbine—our method significantly outperforms LSTM and standard FNO baselines in prediction accuracy and generalization. Crucially, we systematically identify nonlinear intensity as the primary factor constraining FNO performance—a finding previously unreported—and establish a new paradigm for physics-informed, frequency-domain neural operator design.

Technology Category

• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Fourier Neural Operators (FNOs) have emerged as promising surrogates for partial differential equation solvers. In this work, we extensively tested FNOs on a variety of systems with non-linear and non-stationary properties, using a wide range of forcing functions to isolate failure mechanisms. FNOs stand out in modeling linear systems, regardless of complexity, while achieving near-perfect energy preservation and accurate spectral representation for linear dynamics. However, they fail on non-linear systems, where the failure manifests as artificial energy dissipation and manipulated frequency content. This limitation persists regardless of training dataset size, and we discuss the root cause through discretization error analysis. Comparison with LSTM as the baseline shows FNOs are superior for both linear and non-linear systems, independent of the training dataset size. We develop a spectrogram-based loss function that combines time-domain Mean Squared Error (MSE) with frequency-domain magnitude and phase errors, addressing the low-frequency bias of FNOs. This frequency-aware training eliminates artificial dissipation in linear systems and enhances the energy ratios of non-linear systems. The IEA 15MW turbine model validates our findings. Despite hundreds of degrees of freedom, FNO predictions remain accurate because the turbine behaves in a predominantly linear regime. Our findings establish that system non-linearity, rather than dimensionality or complexity, determines the success of FNO. These results provide clear guidelines for practitioners and challenge assumptions about FNOs'universality.

Problem

• Evaluating Fourier Neural Operators' performance on non-linear structural dynamics systems
• Addressing artificial energy dissipation in FNO predictions through spectrogram loss
• Establishing system non-linearity as the key limitation for FNO success

Innovation

• FNOs model linear systems with energy preservation
• Spectrogram loss combines time and frequency domain errors
• Frequency-aware training eliminates artificial energy dissipation