PARA: Patch-based Adaptive multi-Resolution, time-Advancing and scale-equivariant neural PDE solver

We present PARA, a simple yet effective unified framework for advancing multi-resolution states of hyperbolic PDEs. By representing the spatial solution as a tree of independent local patches at varying resolutions (TreeGrid), PARA shifts structural complexity out of the neural network. The future state of each patch is predicted concurrently using a lightweight network from its localized neighborhood (stencil). Restricting the network to the stencil of a patch gives it an inductive bias towards locality while errors fall and flatten once the stencil captures the physical domain of dependence, in contrast to whole-frame models which capture the entire domain yet yield higher errors. Crucially, we introduce scale normalization, which uses the scaling symmetry of the Euler equations to make the solver scale-equivariant by construction, so a single network learns the solution map in dimensionless units. By expanding the TreeGrid into a hierarchy of dense grids, one per resolution level (MultiGrid), PARA cuts every patch with its halo into a stencil of one shape, so all levels advance in a single batched tensor operation. On the Well's challenging compressible Euler benchmark, our symmetry-driven, patch-based approach achieves exceptional accuracy and generalization across spatial resolutions, outperforming whole-frame single-resolution SOTA neural solvers in one-step accuracy and across long-horizon autoregressive rollouts under identical capacity restrictions.