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强大的性能，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FNO在五个性质各异的PDE族中的失效模式：色散型、椭圆型、多尺度流体、金融模型和混沌系统。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、边界或终端条件变化、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于高频模态。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失效模式图谱，并为提升鲁棒性提供了可操作的见解。