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

TL;DR

The paper introduces FINO, a strictly local neural operator for PDEs that replaces fixed finite-difference stencils with learnable kernels and advances solutions via an explicit time-stepping scheme within a U-Net–style encoder–decoder. It provides theoretical guarantees, including a universal approximation result for discrete-time PDE dynamics and a composition error bound that ties one-step accuracy to long-horizon stability. Empirically, FINO delivers higher accuracy and faster training/inference than global, local+global, and transformer baselines across six PDEBench tasks and a climate-modeling dataset, highlighting the value of strict locality for efficiency and fidelity. The work suggests locality-informed neural solvers as a scalable, interpretable, and robust approach for neural PDE surrogates with real-world applicability.

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.

Figures (9)

• Figure 1: FINO Framework. Encoders down-sample and decoders up-sample, with skip connections preserving original feature information.
• Figure 2: Visual comparison of baseline methods and FINO on 1D CNS and 2D Darcyflow.
• Figure 3: Spatiotemporal comparison of LocalFNO and FINO predictions on the 2D Shallow Water equation benchmark. Each row shows model rollouts at successive time steps.
• Figure 4: Comprehensive evaluation of FINO compared to baseline operator networks. (a): Error vs. number of composition blocks for the 1D advection task. (b): Data-scaling performance comparison of FINO and FNO. (c): Training vs. inference time across architectures.
• Figure 5: Ground truth (top) and predicted (bottom) global surface pressure. FINO accurately reconstructs large-scale patterns and regional variations, producing smooth, consistent fields that closely match the reference.

Theorems & Definitions (10)

• Proposition 1: Informal, Local-to-Global Error Bound
• Theorem 2: Universal Approximation of FINO for Discrete Time--Stepped PDE Dynamics
• Lemma 3: Composition Error Estimate
• proof
• Lemma 4
• proof
• Proposition 5: Local-to-Global Error Bound
• proof
• Theorem 6: Universal Approximation of FINO for Discrete Time--Stepped PDE Dynamics
• proof