A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems

Abstract

Partial differential equations (PDEs) on surfaces are ubiquitous in all the nature science. Many traditional mathematical methods has been developed to solve surfaces PDEs. However, almost all of these methods have obvious drawbacks and complicate in general problems. As the fast growth of machine learning area, we show an algorithm by using the physics-informed neural networks (PINNs) to solve surface PDEs. To deal with the surfaces, our algorithm only need a set of points and their corresponding normal, while the traditional methods need a partition or a grid on the surface. This is a big advantage for real computation. A variety of numerical experiments have been shown to verify our algorithm.

Keywords

• PINN
• PDEs on surfaces
• Laplace-Beltrami operator