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

Published: 23 Jun 2025, Last Modified: 23 Jun 2025Greeks in AI 2025 PosterEveryoneRevisionsBibTeXCC BY 4.0
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
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