$\Delta$-PINNs: physics-informed neural networks on complex geometries

TL;DR

This paper introduces a novel eigenfunction-based positional encoding for PINNs, enabling effective solutions on complex geometries where traditional PINNs struggle, thus broadening their applicability to realistic problems.

Contribution

The authors propose a new geometric encoding method for PINNs using Laplace-Beltrami eigenfunctions, enhancing their ability to handle complex shapes.

Findings

Outperforms traditional PINNs on complex geometries

Achieves accurate solutions where traditional PINNs fail

Demonstrates robustness across different physics and shapes

Abstract

Physics-informed neural networks (PINNs) have demonstrated promise in solving forward and inverse problems involving partial differential equations. Despite recent progress on expanding the class of problems that can be tackled by PINNs, most of existing use-cases involve simple geometric domains. To date, there is no clear way to inform PINNs about the topology of the domain where the problem is being solved. In this work, we propose a novel positional encoding mechanism for PINNs based on the eigenfunctions of the Laplace-Beltrami operator. This technique allows to create an input space for the neural network that represents the geometry of a given object. We approximate the eigenfunctions as well as the operators involved in the partial differential equations with finite elements. We extensively test and compare the proposed methodology against traditional PINNs in complex shapes, such as a coil, a heat sink and a bunny, with different physics, such as the Eikonal equation and heat transfer. We also study the sensitivity of our method to the number of eigenfunctions used, as well as the discretization used for the eigenfunctions and the underlying operators. Our results show excellent agreement with the ground truth data in cases where traditional PINNs fail to produce a meaningful solution. We envision this new technique will expand the effectiveness of PINNs to more realistic applications.