Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networks
This addresses the curse of dimensionality in solving PDEs for minimal surfaces, offering a computationally efficient alternative to classical methods, but it appears incremental as it applies an existing PINN method to this specific problem.
The paper tackles approximating minimal surfaces in high dimensions using Physics-Informed Neural Networks (PINNs), achieving scalability to higher dimensions with training on a laptop without GPU, though specific numerical results are not provided.
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.