We introduce an approach for solving PDEs over manifolds using physics informed neural networks whose architecture aligns with spectral methods. The networks are trained to take in as input samples of an initial condition, a time stamp and point(s) on the manifold and then output the solution's value at the given time and point(s). We provide proofs of our method for the heat equation on the interval and examples of unique network architectures that are adapted to nonlinear equations on the sphere and the torus. We also show that our spectral-inspired neural network architectures outperform the standard physics informed architectures. Our extensive experimental results include generalization studies where the testing dataset of initial conditions is randomly sampled from a significantly larger space than the training set.

翻译：我们提出了一种利用物理信息神经网络求解流形上偏微分方程的方法，该网络架构与谱方法相统一。网络训练以初始条件样本、时间戳和流形上点(们)为输入，输出给定时刻与该点(们)的解值。我们给出了区间上热方程方法的数学证明，并展示了适用于球面和环面上非线性方程的唯一网络架构实例。同时证明我们的谱启发式神经网络架构优于标准物理信息架构。大量实验结果包含泛化研究：测试集初始条件从远大于训练集的样本空间中随机采样。