Preserving Local Detail in Neural PDE Simulators with Structured Latent States

High-fidelity numerical simulation of complex physical systems often incurs prohibitive computational costs, particularly for time-dependent problems that require repeated fine-resolution updates. Neural PDE simulators provide a data-driven alternative, while latent simulators further reduce rollout cost by evolving a spatially compressed representation of the physical field. However, strong spatial compression can remove the localized structures that require fine resolution in the first place. In this paper, we propose a neural PDE simulation approach that preserves localized fine-scale structure under strong spatial compression. Motivated by classical numerical methods, the approach emphasizes expressive local representations through two design principles: (i) a spatially structured latent state that preserves where information occurs in the physical domain, and (ii) a partition-of-unity blending scheme that reconstructs a continuous physical field from local latent representations. We instantiate this approach in Spatial Patch Neural Simulator (SPANS), an efficient latent simulator for high-resolution, time-dependent PDEs. Its architecture couples short- and long-range attention processor with an expressive decoder whose patch-conditioned weights vary with spatial position. Across seven time-dependent simulation problems spanning structured grids, irregular meshes, and point clouds, SPANS achieves strong rollout accuracy, preserving localized spatial detail even under substantial compression, while also maintaining lower rollout time than most methods.