Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation
Summary
This paper investigates the generalization behavior of Fourier Neural Operators and Deep Operator Networks under distribution shifts in a variable-coefficient wave equation, revealing that FNO struggles with high-frequency inputs while DeepONet shows milder degradation.
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Paper page - Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation
Source: https://huggingface.co/papers/2605.12997 Published on May 13
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Submitted byhttps://huggingface.co/lainmn
An Luoon May 14
Abstract
Neural operators for PDE solving show different generalization behaviors under distribution shifts, with Fourier Neural Operators and Deep Operator Networks exhibiting distinct responses to smoothness and frequency variations.
Neural operatorslearn to map initial conditions to the terminal solution ofpartial differential equations(PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structureddistribution shiftsremains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, theFourier Neural Operator(FNO) and theDeep Operator Network(DeepONet). To examine their generalization underdistribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations infrequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization underdistribution shiftsin operator learning, underscoring the role of architecturalrepresentation biasin developing more reliableneural operatorsfor physics-based PDE simulations beyond the training distribution.
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