Solver and Surrogate Choices for PDE Inference and Forecasting: A Paired Benchmark with Observation Prefixes

Abstract Partial differential equation (PDE) forecasting from sparse noisy observations requires parameter infer ence and future propagation. We compare pipelines using surrogate inference and forecasting (A0), surrogate inference followed by a numerical forecast (A1), and numerical prefix inference followed by a numerical forecast (P). The main benchmark covers three one-dimensional periodic PDE families, Fourier neural operator (FNO) models and U-Nets, three matched training seeds, and 200 test initial condi tions per model. A1 lowers error in all 12 Allen–Cahn and Burgers configurations. A separate evaluation against a refined numerical reference reproduces this direction in all 12 configurations, with paired 95% intervals below zero. Increasing FNO training from 24 to 96 initial conditions at fixed epochs improves surrogate accuracy and narrows the additional gain from numerical forecasting. At three observed out puts, mean A1 minus A0 error changes from −0.03295 to −0.00687 for Allen–Cahn and from −0.00478 to −0.00175 for Burgers. Complete-query measurements also show substantial dependence on imple mentation: restricting candidate rollout to the observed prefix preserves A1 forecasts while reducing three-output query time by about 27% for Burgers/FNO and 48% for Burgers/U-Net. Direct Fourier propagation makes numerical inference cheaper than surrogate-assisted inference across the advection diffusion horizon grid. At the operational likelihood temperatures, its forecast is more accurate at eight observed outputs and less accurate at three. Development-selected temperatures also change the P–A1 prefix comparison on a separate test block. These experiments quantify how surrogate training, infer ence calibration, candidate rollout length and numerical propagation affect the accuracy and cost of inference-to-forecast pipelines. MSC Classification: 65M32 , 65Y20 , 68T07