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This research addresses a critical validation gap in physics-informed machine learning by demonstrating that neural PDE surrogates often merely memorize the discretization errors of their training solvers rather than learning the underlying physical laws. The authors introduce a Fourier-symbol diagnostic to empirically distinguish between fidelity to exact evolution and imitation of numerical schemes, revealing near-perfect replication of solver artifacts in linear advection and Burgers dynamics. This work is essential for establishing rigorous benchmarks that ensure AI models generalize beyond specific numerical implementations.
The Symbol of the Surrogate: Measuring Numerical Provenance in Neural PDE Solvers
arXiv:2610.09255v1 Announce Type: new 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.
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.
Machine Learning (cs.LG); Analysis of PDEs (math.AP); Machine Learning (stat.ML)
arXiv:2610.09255 [cs.LG]
(or arXiv:2610.09255v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.09255 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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