M-PINNs: Manifold - Physics-Informed Neural Networks for Solving PDEs on Curved and Changing Domains

Riccardo Valperga, Samuele Papa, Stratis Gavves

Keywords: PINNs, geometry-informed neural networks, PDEs, Neural Operators, Manifold Learning

TL;DR: We propose a neural operator for learning local manifold embeddings to solve PDEs on complex shapes using PINNs

Abstract:

Solving partial differential equations on domains with complex geometries is a
significant challenge. Physics-Informed Neural Networks (PINNs) would offer a
promising mesh-free approach, but their application to manifolds is limited. This
paper introduces Manifold Physics-Informed Neural Networks ($\mathcal{M}$-PINNs) that
leverage domain decomposition for partitioning complex shapes into simpler charts,
for which we learn local embeddings. The core of our method is a novel universal
autoencoder architecture designed to generate these embeddings zero-shot for any
given chart. By training this system on a diverse set of charts, the decoders for new,
unseen charts can be obtained without retraining or adaptation. These decoders
then serve as the local embeddings within which separate PINNs are trained to
solve the PDE, effectively integrating the domain’s geometry into the process.
This approach results in a scalable and adaptable framework for solving PDEs on
complex manifolds.

Submission Number: 167