PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils

Chun-Wun Cheng, Bin Dong, Carola-Bibiane Schönlieb, Angelica I Aviles-Rivero

Keywords: Neural Operator, PDEs, AI for Science

Abstract: Neural operator models for solving partial differential equations (PDEs) often rely on global mixing mechanisms—such as spectral convolutions or attention—which tend to oversmooth sharp local dynamics and introduce high computational cost. We present FINO, a finite-difference–inspired neural architecture that enforces strict locality while retaining multiscale representational power. FINO replaces fixed finite-difference stencil coefficients with learnable convolutional kernels and evolves states via an explicit, learnable time-stepping scheme. A central Local Operator Block leverage a differential stencil layer, a gating mask, and a linear fuse step to construct adaptive derivative-like local features that propagate forward in time. Embedded in an encoder–decoder with a bottleneck, FINO captures fine-grained local structures while preserving interpretability. We establish (i) a composition error bound linking one-step approximation error to stable long-horizon rollouts under a Lipschitz condition, and (ii) a universal approximation theorem for discrete time-stepped PDE dynamics. (iii) Across six benchmarks and a climate modelling task, FINO achieves up to 44\% lower error and up to around 2× speedups over state-of-the-art operator-learning baselines, demonstrating that strict locality with learnable time-stepping yields an accurate and scalable foundation for neural PDE solvers.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 12223