发表机构
Indian Institute of Technology Gandhinagar（印度理工学院甘地讷格尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出傅里叶符号诊断方法，通过比较代理模型与训练方案的算子特征，揭示神经PDE代理模型可能模仿数值方案而非精确演化，其差异可达99.8%以上。
AI 中文摘要
神经PDE代理模型在数值求解器输出上进行训练，这些输出既包含物理演化，也包含求解器特定的离散化误差。由于代理模型同样针对来自同一求解器的留出轨迹进行评估，标准基准无法区分对精确演化的忠实度与对数值方案的模仿。我们引入了一种经验傅里叶符号诊断方法，该方法用单个傅里叶模式探测训练后代理模型的线性化单步算子，并将其与精确演化和训练方案参考进行比较。为了解决架构频谱偏差，我们在具有正交耗散和色散特征的方案上训练相同网络，并比较它们学习到的算子。在线性平流中，学习到的代理模型再现了训练方案的振幅和相位误差，双方案差异达到解析预测的全模仿上限的99.8%以上。同样的行为也出现在非局部傅里叶神经算子中，并在算子层面出现在非线性Burgers动力学中。这些结果表明，与求解器生成的测试数据的一致性本身并不能确立对精确演化的忠实度。傅里叶符号测量提供了数值来源的直接诊断。
英文摘要
Neural PDE surrogates are trained on numerical solver outputs that contain both physical evolution and solver-specific discretization errors. Because surrogates are also evaluated against held-out trajectories from the same solver, standard benchmarks cannot distinguish fidelity to the exact evolution from imitation of the numerical scheme. We introduce an empirical Fourier-symbol diagnostic that probes a trained surrogate's linearized one-step operator with individual Fourier modes and compares it with both exact-evolution and training-scheme references. To address architectural spectral bias, we train identical networks on schemes with orthogonal dissipative and dispersive signatures and compare their learned operators. In linear advection, the learned surrogates reproduce the training schemes' amplitude and phase errors, with the twin-scheme difference reaching more than 99.8\% of the analytically predicted full-imitation ceiling. The same behavior occurs for a non-local Fourier neural operator and at the operator level for nonlinear Burgers dynamics. These results show that agreement with solver-generated test data does not by itself establish fidelity to the exact evolution. Fourier-symbol measurements provide a direct diagnostic of numerical provenance.
CommentsAccepted at NeurIPS 2026 AI for Science Workshop
