Spatial Context or Memory? Sizing the Inputs of Neural Surrogates for Partially Observed Fields

Ruige Kong

Neural surrogates that forecast a physical field from sparse, noisy sensors read a window of nearby sensors and a history of past snapshots. Practitioners choose both by comparing fitted models with each other. Such comparisons cannot tell a model's shortfall from missing information. We measure the trade-off between window and history against the Bayes-optimal error of the same readings. An exact Gaussian analysis shows that noisy or sparse observation of a field with local dynamics makes the optimal forecast read distant sensors only through spatial correlation. On a forced linear field with a known optimum, trained models that read only the window come within 1% of it. We then train such models on viscous Burgers flow and the chaotic Kuramoto-Sivashinsky (KS) equation, with every deciding comparison fixed before any test evaluation. In every field except KS, eight snapshots remove a smaller fraction of the one-snapshot error as the lead time, the number of steps to the forecast target, grows. In KS, for windows up to one correlation length, they remove a larger fraction. This lead-time inversion persists across sensor densities, three architectures and a two-dimensional…