HybridOSS: Orthogonal Scale Splitting for Hybrid Differentiable PDE Solvers

Fine-resolution simulation of partial differential equations (PDEs) is important for resolving small-scale structures and cross-scale interactions, but becomes computationally expensive over long time horizons. Neural surrogates reduce this cost but can suffer from error accumulation, while hybrid models typically require the numerical solver and corrected state to operate at the same spatial resolution. We propose HybridOSS, an orthogonal scale splitting framework that decouples numerical-solver resolution from state resolution. HybridOSS decomposes the fine-resolution state into complementary resolved and unresolved components. A differentiable numerical solver advances the resolved state on a coarse grid, while a closure module corrects the resolved dynamics and a refinement module explicitly evolves the unresolved state. Complementary spectral projections constrain the learned updates to orthogonal subspaces while preserving cross-scale coupling. Our instantiated HybridOSS model reduces global prediction error by 31% and 69% over the strongest hybrid baseline on KS and Allen–Cahn, respectively, while improving both resolved- and unresolved-scale accuracy. On differentiable OpenFOAM for 3D incompressible Navier–Stokes turbulence, HybridOSS maintains lower error over long autoregressive rollouts and nearby unseen viscosity values while achieving approximately a $19\times$ inference speedup over the fine-grid solver.