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，其网络架构与谱方法相匹配。该网络经过训练后，能够以初始条件样本、时间戳及流形上的点作为输入，并输出给定时刻与位置处的解值。我们提供了该方法在一维区间热方程上的理论证明，并展示了针对球面与环面上非线性方程所设计的独特网络架构实例。研究还表明，受谱方法启发的神经网络架构优于标准物理信息架构。大量实验包含了泛化性研究：测试数据集中的初始条件从远大于训练集的随机空间中进行采样。