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.

翻译：傅里叶神经算子（FNOs）在学习偏微分方程（PDEs）的解映射方面表现出色，但其在分布偏移、长时程推演和结构扰动下的鲁棒性仍鲜为人知。我们提出了一个系统化的压力测试框架，用于探究FNOs在五个性质各异的PDE族中的故障模式：色散型、椭圆型、多尺度流体、金融和混沌系统。我们并非优化分布内精度，而是设计了受控的压力测试——包括参数偏移、边界或终端条件改变、结合谱分析的分辨率外推以及迭代推演——以揭示诸如谱偏差、积分误差累积以及对受限边界机制的过拟合等脆弱性。我们的大规模评估（1,000个训练模型）表明，参数或边界条件的分布偏移可使误差增加一个数量级以上，而分辨率变化主要将误差集中在高频模态。输入扰动通常不会放大误差，但最坏情况（例如局部泊松扰动）仍然具有挑战性。这些发现为算子学习提供了一个可比较的故障模式图谱，并为提升鲁棒性提供了可操作的见解。