OPERATOR LEARNING ON FREE-FORM GEOMETRIES

Louis Serrano, Jean-Noël Vittaut, patrick gallinari

Keywords: Partial Differential Equations, PDEs, Implicit Neural Representations, INRs, continuous models, generalization, physics, Operator Learning, Initial Value Problem, Geometric Design

TL;DR: We introduce CORAL a new model for Operator Learning without constraints on training mesh or input sampling, and validate its performance on Initial Value Problems and a Geometric Design Task.

Abstract: Operator Learning models usually rely on a fixed sampling scheme for training which might limit their ability to generalize to new situations. We present CORAL, a new method which leverages Coordinate-Based Networks for OpeRAtor Learning without any constraints on the training mesh or input sampling. CORAL is able to solve complex Initial Value Problems such as 2D Navier-Stokes or 3D spherical Shallow-Water and can perform zero-shot super-resolution to recover a dense grid, even when the training grid is irregular and sparse. It can also be applied to the task of geometric design with structured or point-cloud data, to infer the steady physical state of a system given the characteristics of the domain.