测试神经算子在多种微分方程上的鲁棒性，发现精度高不等于抗干扰强。
Diagnosing Failure Modes of Neural Operators Across Diverse PDE Families
- 设计标准化压力测试框架，模拟真实部署中的参数与边界变化
- 750个模型测试显示，分布内精度无法预测泛化能力下降
- 不同架构对不同方程类型表现差异大，需针对性评估
神经微分方程求解器作为多类偏微分方程的可学习代理模型，其核心挑战不仅是固定分布上的插值，更在于系数、边界条件、离散化和滚动时间跨度等结构化变化下的泛化能力。然而当前评估仍以分布内测试误差为主，难以衡量鲁棒性。本文提出一种面向实际部署场景的标准化压力测试框架，并在三种代表性架构（FNO、DeepONet类模型、CNO）上，针对五类差异显著的PDE家族（色散型、椭圆型、多尺度流体、金融、混沌系统）进行验证。通过750个训练模型，采用基线归一化退化因子结合谱分析与滚动诊断，揭示了强分布内精度无法可靠预测鲁棒性，且失效模式同时依赖于架构与方程类型。研究结果为神经PDE求解器的鲁棒性评估提供了更清晰基准，并建议将函数空间中结构化扰动下的泛化能力作为首要评估目标。
原文摘要 · Abstract (English)
Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under structured shifts in coefficients, boundary conditions, discretization, and rollout horizon. Yet evaluation is still often dominated by in-distribution test error, making robustness difficult to assess. We introduce a standardized stress-testing framework for neural PDE solvers under deployment-relevant shift. We instantiate it on three representative architectures -- Fourier Neural Operators (FNOs), a DeepONet-style model, and convolutional neural operators (CNOs) -- across five qualitatively different PDE families: dispersive, elliptic, multi-scale fluid, financial, and chaotic systems. Across 750 trained models, we measure robustness using baseline-normalized degradation factors together with spectral and rollout diagnostics. The resulting comparisons reveal that strong in-distribution accuracy does not reliably predict robustness, and that failure patterns depend jointly on architecture and PDE family. Our results provide a clearer basis for evaluating robustness claims in neural PDE solvers and suggest that function-space generalization under structured shift should be treated as a first-class evaluation target.
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