Better Simulations Do Not Guarantee Better Surrogates: A Paired Study of Numerical Fidelity

Numerical simulations approximate PDE solutions, but their usefulness as training data also depends on the dynamics learned from them. We investigate this distinction with PDE-FIDELITY, a paired benchmark that varies the polynomial degree of one, three, and five in the discontinuous Galerkin (DG) method while keeping physical instances and the observation interface fixed. This design separates numerical label accuracy, prediction from common reference histories, and autonomous rollout. Across four systems and seven architectures, degree-five supervision improves label accuracy and aggregate one-step prediction over degree one, yet the rollout benefit is system-dependent and can reverse. In GLM–MHD, the aggregate degree-five/degree-one rollout error ratio is roughly three; the reversal persists under the tested training controls and is confirmed for U-Net on fresh physical instances. An exact DG construction shows that such a reversal can arise even at the unique training-loss minimizers. Within this construction, enforcing mean preservation removes the reversal without adding learned parameters, demonstrating how model structure can change the value of a numerical source. Finite-horizon propagation yields conditional vector-error bounds that retain defect directions and temporal interactions. Corresponding reconstruction experiments show close agreement with neural rollout errors over short windows, with model-dependent limits at longer horizons. Complementary finer-grid experiments also produce substantial U-Net gains, showing that accurate supervision can help without implying a universal source ranking. These results motivate evaluating numerical supervision through the autonomous dynamics learned from it, rather than inferring its value from label or one-step accuracy alone.