Partition of Unity Physics-Informed Neural Networks (POU-PINNs): An Unsupervised Framework for Physics-Informed Domain Decomposition and Mixtures of Experts

TL;DR

This paper introduces an unsupervised framework using partition of unity networks to automatically decompose domains and identify physics in PDEs, enhancing the accuracy of physics-informed neural networks for complex problems.

Contribution

It proposes a novel unsupervised domain decomposition method with physics residual-based loss, enabling discovery of spatial subdomains and physics parameters without labeled data.

Findings

Successfully applied to porous media thermal ablation.

Effective in ice-sheet modeling.

Improves accuracy by dividing solution space into subdomains.

Abstract

Physics-informed neural networks (PINNs) commonly address ill-posed inverse problems by uncovering unknown physics. This study presents a novel unsupervised learning framework that identifies spatial subdomains with specific governing physics. It uses the partition of unity networks (POUs) to divide the space into subdomains, assigning unique nonlinear model parameters to each, which are integrated into the physics model. A vital feature of this method is a physics residual-based loss function that detects variations in physical properties without requiring labeled data. This approach enables the discovery of spatial decompositions and nonlinear parameters in partial differential equations (PDEs), optimizing the solution space by dividing it into subdomains and improving accuracy. Its effectiveness is demonstrated through applications in porous media thermal ablation and ice-sheet modeling, showcasing its potential for tackling real-world physics challenges.