Physics-informed neural networks for transformed geometries and manifolds

TL;DR

This paper introduces a method to extend physics-informed neural networks (PINNs) to handle complex and varying geometries by integrating geometric transformations, enabling applications on manifolds and shape optimization.

Contribution

The authors propose a novel approach that incorporates diffeomorphisms into PINNs, allowing for robust modeling on deformed domains and manifolds, and facilitating shape optimization.

Findings

Effective on Eikonal equation on spiral

Successful on Poisson problem on surfaces

Demonstrated shape optimization with Laplace operator

Abstract

Physics-informed neural networks (PINNs) effectively embed physical principles into machine learning, but often struggle with complex or alternating geometries. We propose a novel method for integrating geometric transformations within PINNs to robustly accommodate geometric variations. Our method incorporates a diffeomorphism as a mapping of a reference domain and adapts the derivative computation of the physics-informed loss function. This generalizes the applicability of PINNs not only to smoothly deformed domains, but also to lower-dimensional manifolds and allows for direct shape optimization while training the network. We demonstrate the effectivity of our approach on several problems: (i) Eikonal equation on Archimedean spiral, (ii) Poisson problem on surface manifold, (iii) Incompressible Stokes flow in deformed tube, and (iv) Shape optimization with Laplace operator. Through these examples, we demonstrate the enhanced flexibility over traditional PINNs, especially under geometric variations. The proposed framework presents an outlook for training deep neural operators over parametrized geometries, paving the way for advanced modeling with PDEs on complex geometries in science and engineering.