Hyperbolic mean curvature flow computed by physics-informed neural networks

TL;DR

This paper introduces a mesh-free physics-informed neural network approach to simulate hyperbolic mean curvature flow of curves and surfaces, enabling efficient high-dimensional PDE solutions without discretization.

Contribution

It is the first to apply PINNs to hyperbolic geometric evolution equations, offering a novel numerical analysis method for these complex PDEs.

Findings

Successful numerical simulations with diverse initial conditions

Elimination of discretization and meshing in PDE solving

Effective handling of high-dimensional hyperbolic PDEs

Abstract

In this paper, we explore the evolution of plane curves and surfaces governed by the hyperbolic mean curvature flow. We propose a mesh-free approach based on the physics-informed neural networks (PINNs), which eliminates the need for discretization and meshing of computational domains, and is efficient in solving partial differential equations involving high dimensions. To the best of our knowledge, this is the first result on the numerical analysis by employing the PINNs for the hyperbolic geometric evolution equations in the literature. The effectiveness of this method is demonstrated through several numerical simulations by selecting diverse initial curves and surfaces, as well as both constant and non-constant initial velocities.