Operator Learning with Neural Fields: Tackling PDEs on General Geometries

• URL: http://arxiv.org/abs/2306.07266v2
• Date: Thu, 30 Nov 2023 13:59:53 GMT
• Title: Operator Learning with Neural Fields: Tackling PDEs on General Geometries
• Authors: Louis Serrano, Lise Le Boudec, Armand Kassa\"i Koupa\"i, Thomas X Wang, Yuan Yin, Jean-No\"el Vittaut, Patrick Gallinari
• Abstract summary: Machine learning approaches for solving partial differential equations require learning mappings between function spaces. New CORAL method leverages coordinate-based networks for PDEs on some general constraints.
• Score: 15.65577053925333
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

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