Forcing and Diagnosing Failure Modes of Fourier Neural Operators Across Diverse PDE Families

• URL: http://arxiv.org/abs/2601.11428v1
• Date: Fri, 16 Jan 2026 16:47:44 GMT
• Title: Forcing and Diagnosing Failure Modes of Fourier Neural Operators Across Diverse PDE Families
• Abstract summary: Fourier Neural Operators (FNOs) have shown strong performance in learning solution maps of partial differential equations (PDEs)<n>We present a systematic stress-testing framework that probes failure modes of FNOs across five qualitatively different PDE families.<n>These findings provide a comparative failure-mode atlas and actionable insights for improving robustness in operator learning.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Fourier Neural Operators (FNOs) have shown strong performance in learning solution maps of partial differential equations (PDEs), but their robustness under distribution shifts, long-horizon rollouts, and structural perturbations remains poorly understood. We present a systematic stress-testing framework that probes failure modes of FNOs across five qualitatively different PDE families: dispersive, elliptic, multi-scale fluid, financial, and chaotic systems. Rather than optimizing in-distribution accuracy, we design controlled stress tests--including parameter shifts, boundary or terminal condition changes, resolution extrapolation with spectral analysis, and iterative rollouts--to expose vulnerabilities such as spectral bias, compounding integration errors, and overfitting to restricted boundary regimes. Our large-scale evaluation (1{,}000 trained models) reveals that distribution shifts in parameters or boundary conditions can inflate errors by more than an order of magnitude, while resolution changes primarily concentrate error in high-frequency modes. Input perturbations generally do not amplify error, though worst-case scenarios (e.g., localized Poisson perturbations) remain challenging. These findings provide a comparative failure-mode atlas and actionable insights for improving robustness in operator learning.

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