arXiv：cs.LG· Ridham Patel·· 4 小时前AI 评分46
神经PDE代理模型的符号溯源：用Fourier符号诊断数值来源
The Symbol of the Surrogate: Measuring Numerical Provenance in Neural PDE Solvers
AI 导读
研究提出一种经验性Fourier符号诊断方法，用单个Fourier模态探测已训练神经PDE代理模型的线性化单步算子，并与精确演化及训练格式参考对比。在线性平流中，学到的代理模型复现了训练格式的振幅与相位误差，双格式差异达到解析预测完全模仿上限的99.8%以上；该现象在非局部Fourier神经算子和非线性Burgers动力学中同样出现。结果表明，与求解器生成测试数据的一致并不能证明对精确演化的保真。
正文
Abstract: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.
| Comments: | Accepted at NeurIPS 2026 AI for Science Workshop |
| Subjects: | Machine Learning (cs.LG); Analysis of PDEs (math.AP); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.09255 [cs.LG] |
| (or arXiv:2610.09255v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.09255 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Ridham Patel [view email]
[v1]
Wed, 7 Oct 2026 00:42:11 UTC (266 KB)
来源：arXiv：cs.LG · arxiv.org