Numerical Methods For PDEs Over Manifolds Using Spectral Physics Informed Neural Networks
This work addresses solving PDEs on complex geometries for computational science and engineering, presenting an incremental improvement through spectral-inspired neural network designs.
The authors tackled solving PDEs on manifolds by introducing spectral physics-informed neural networks, demonstrating that these architectures outperform standard physics-informed methods in experiments, including generalization tests with randomly sampled initial conditions from a larger space.
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.