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（一种重要的偏微分方程）在高维情况下的数值近似。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会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。直到最近几年，机器学习研究者才得以缓解这一问题。本文所采用的解法模型称为物理信息神经网络（PINN），该模型训练深度神经网络（DNN）求解极小曲面偏微分方程。该方法可扩展至高维情形，并且即便在无GPU的笔记本电脑上也可相对快速地完成训练。由于无法直接观察高维输出，我们将数据呈现为固定足够多轴的高维形状片段，使其可通过三维图形观察。本文不仅测试了该方法的功能性，还将探索其可能存在的性能局限性。