Implicit Neural Representation for PDEs

Topic: Operator Learning with Neural Fields for Partial Differential Equations

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.

Speaker(s):

Louis Serrano is a PhD candidate at Sorbonne Universite in Paris. Previously he completed his MSc in Statistics at Oxford University. His current interests include operator learning, implicit neural representations, and machine learning for physics and science in general.