这篇论文给做神经算子、物理信息学习或科学计算的团队提供了一个关键诊断工具——预测误差可能骗人，但雅可比谱审计能揪出模型学没学到真正的物理机制。做PDE代理模型或算子学习的建议点开看看，能帮你避免模型“看起来准、用起来崩”的坑。
现有神经算子评估主要依赖预测误差，但准确输出不代表模型学到了正确的局部动力学结构。研究者提出一种基于雅可比矩阵的谱审计方法，通过将网络输出对查询函数求导，得到学习到的切向算子，再投影到傅里叶模式上，揭示频率依赖增益、相位结构和跨模式耦合等局部谱特征。该方法在多个基准测试中发现了预测误差无法暴露的问题，如高频退化、错误相位恢复和提示-算子不一致。结果表明，预测精度和局部算子保真度是神经算子的两个独立属性，该框架可用于稳定性、敏感性和算子一致性的诊断。
Spectral Audit of In-Context Operator Networks
Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.