The Symbol of the Surrogate: Measuring Numerical Provenance in Neural PDE Solvers

Key points

• 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.1
• 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.1
• To address architectural spectral bias, we train identical networks on schemes with orthogonal dissipative and dispersive signatures and compare their learned operators.1
• The same behavior occurs for a non-local Fourier neural operator and at the operator level for nonlinear Burgers dynamics.1

Sources (1)

• [1]The Symbol of the Surrogate: Measuring Numerical Provenance in Neural PDE Solvers
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 7, 12:42 AM

  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.

Extractive summary: sentences quoted from the sources.