PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Cheng, Chun-Wun Dong, Bin Schönlieb, Carola-Bibiane Aviles-Rivero, Angelica I |
| author_facet | Cheng, Chun-Wun Dong, Bin Schönlieb, Carola-Bibiane Aviles-Rivero, Angelica I |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26186 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils Cheng, Chun-Wun Dong, Bin Schönlieb, Carola-Bibiane Aviles-Rivero, Angelica I Machine Learning Numerical Analysis 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. |
| title | PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2509.26186 |