# Numerical Methods For PDEs Over Manifolds Using Spectral Physics Informed Neural Networks

arXiv: https://arxiv.org/abs/2302.05322 · HTML: https://mlsanity.com/paper/2302.05322/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 27 references that are in the map.
- First posted on arXiv 10 Feb 2023.

## Builds on (most cited first, 6 of 6)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems](https://mlsanity.com/paper/1904.05417/) (2019)
- [Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems](https://mlsanity.com/paper/2202.02710/) (2022)

## Built on by (most cited first, 1 of 1)

- [Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks](https://mlsanity.com/paper/2308.09605/) (2023)

## Abstract (arXiv)

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.