Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networks

TL;DR

This paper demonstrates the use of Physics-Informed Neural Networks to approximate high-dimensional minimal surfaces, overcoming classical computational limitations and enabling scalable solutions in higher dimensions.

Contribution

It introduces a scalable PINN-based approach for solving high-dimensional minimal surface PDEs, addressing the curse of dimensionality in traditional methods.

Findings

PINNs can effectively approximate high-dimensional minimal surfaces.

The method is computationally feasible on standard hardware.

Potential limitations of PINNs in high-dimensional PDEs are discussed.

Abstract

In this paper, we compute numerical approximations of the minimal surfaces, an essential type of Partial Differential Equation (PDE), in higher dimensions. Classical methods cannot handle it in this case because of the Curse of Dimensionality, where the computational cost of these methods increases exponentially fast in response to higher problem dimensions, far beyond the computing capacity of any modern supercomputers. Only in the past few years have machine learning researchers been able to mitigate this problem. The solution method chosen here is a model known as a Physics-Informed Neural Network (PINN) which trains a deep neural network (DNN) to solve the minimal surface PDE. It can be scaled up into higher dimensions and trained relatively quickly even on a laptop with no GPU. Due to the inability to view the high-dimension output, our data is presented as snippets of a higher-dimension shape with enough fixed axes so that it is viewable with 3-D graphs. Not only will the functionality of this method be tested, but we will also explore potential limitations in the method's performance.