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

TL;DR

This study addresses the robustness of Fourier Neural Operators (FNOs) when faced with real-world distribution shifts, long-horizon rollouts, and structural perturbations across five PDE families, rather than optimizing in-distribution accuracy. It introduces a stress-testing framework with parameter and boundary shifts, resolution extrapolation, rollouts, and input perturbations, evaluated on an ensemble of $N=200$ seeds per PDE to yield a degradation factor $D$ that quantifies performance loss. Across nonlinear Schrödinger, Poisson, Navier–Stokes, Black–Scholes, and Kuramoto–Sivashinsky equations, the authors observe universal failure modes such as spectral bias toward high-frequency content, large errors under boundary/parameter shifts, and substantial long-horizon rollout degradation, alongside PDE-specific vulnerabilities (e.g., discontinuous payoffs in Black–Scholes). The findings highlight the importance of multi-scale, boundary-aware, and stability-promoting architectures, and lay out a practical robustness evaluation suite to benchmark and improve operator learning for reliable, real-world PDE solving. These insights advance the understanding of where neural operators can be trusted and how to design models that better generalize under realistic perturbations.

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.

Figures (8)

• Figure 1: Mean error degradation factors across PDE families and stress tests. Each entry reports the mean out-of-distribution error normalized by the in-distribution baseline, aggregated across random seeds. Darker colors indicate more severe failure. White cells denote stress tests not applicable to a given PDE.
• Figure 2: Degradation factors for the nonlinear Schrödinger equation. Extrapolation to stronger nonlinearity ($\kappa$) causes the largest performance drop, while resolution extrapolation remains benign consistent with spectral bias.
• Figure 3: Fourier-domain distribution of prediction error for the nonlinear Schrödinger equation under resolution extrapolation. Error energy is concentrated at high frequencies beyond the training bandwidth, indicating spectral bias.
• Figure 4: Degradation factors for the Poisson equation. Resolution extrapolation and boundary condition shifts cause order-of-magnitude error increases, while coefficient scaling alone is comparatively benign.
• Figure 5: Error degradation for 2D Navier--Stokes across stress tests. Long-horizon rollouts induce substantial error amplification, whereas parameter, resolution, and perturbation shifts produce only mild degradation in global $L^2$ error.