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：我们提出了一种基于物理信息神经网络的流形上偏微分方程求解方法，其网络架构与谱方法相兼容。该网络通过训练，能够以初始条件采样值、时间戳和流形上的点（或多点）作为输入，输出给定时刻和位置的解值。我们针对区间上的热传导方程给出了方法证明，并展示了适用于球面与环面上非线性方程的独特网络架构实例。研究结果表明，受谱方法启发的神经网络架构在性能上优于标准的物理信息架构。我们开展了大量实验，其中包括泛化能力研究：测试数据集中的初始条件从远大于训练集的样本空间中随机采样生成。