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.

翻译：本文计算了极小曲面（偏微分方程的一种重要类型）在高维情形下的数值近似解。由于维数灾难，经典方法无法处理此类问题——这些方法的计算成本随问题维度的增加呈指数级增长，远超任何现代超级计算机的计算能力。仅在过去几年中，机器学习研究者才得以缓解这一问题。本文采用的求解方法是一种称为物理信息神经网络（PINN）的模型，它通过训练深度神经网络（DNN）来求解极小曲面偏微分方程。该方法可扩展至高维情形，即便在无GPU的笔记本电脑上也能相对快速地训练。由于无法直接可视化高维输出，我们将数据呈现为固定足够数量坐标轴后的高维形状片段，使其可通过三维图形展示。本文不仅测试了该方法的有效性，还将探讨其性能的潜在局限性。