Track papers across labs, topics, and trends.
Search papers, labs, and topics across Lattice.
This paper introduces an empirical Fourier-symbol diagnostic to disentangle true physical fidelity from numerical scheme imitation in neural PDE surrogates trained on discretized solver outputs. By comparing the linearized one-step operator of a surrogate against exact evolution and training scheme references across individual Fourier modes, the authors reveal that current benchmarks conflate these two aspects. Results demonstrate that neural surrogates often learn to perfectly replicate the amplitude and phase errors of the training schemes, indicating a lack of fidelity to the underlying physical evolution.
Neural PDE solvers are not learning physics, but rather the specific discretization errors of their training solvers, failing to generalize beyond the training scheme's spectral signature.
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.