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

• URL: http://arxiv.org/abs/2509.26186v1
• Date: Tue, 30 Sep 2025 12:42:32 GMT
• Title: PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils
• Authors: Chun-Wun Cheng, Bin Dong, Carola-Bibiane Schönlieb, Angelica I Aviles-Rivero,
• Abstract summary: We present FINO, a finite-difference-inspired neural architecture that enforces strict locality.<n>FINO replaces fixed finite-difference stencil coefficients with learnable convolutional kernels.<n>It achieves up to 44% lower error and up to around 2times speedups over state-of-the-art operator-learning baselines.
• Score: 20.49015396991881
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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\times 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.

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