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

TL;DR

This work addresses ill-posed inverse problems in PDEs with strong spatial variability by introducing Partition of Unity Physics-Informed Neural Networks (POU-PINNs), which combine Partition of Unity Networks with physics-informed neural networks to discover spatial subdomains and local conductivities in an unsupervised manner. Conductivity is modeled as $K_{POU}(x)=\sum_i \varphi_i(x) e^{c_i}$ with $\sum_i \varphi_i(x)=1$, and embedded into the Darcy diffusion equation $\nabla \cdot(-K\nabla u)=f$, enabling localized physics to be learned across subdomains while enforcing flux continuity at interfaces. The authors demonstrate forward and inverse capabilities on porous media thermal ablation and ice-sheet-like problems using manufactured solutions and various BCs, achieving improved accuracy and convergence through domain decomposition. Overall, POU-PINNs offer a scalable framework for solving real-world PDE challenges with spatially varying properties by unsupervisedly partitioning the domain and learning localized physics, potentially benefiting applications in geophysics, thermal transport, and glaciology.

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.

Figures (23)

• Figure : Fig.1 Physics-informed neural network: In our cases, we consider only the input dimensions, the partial differential equation, and boundary conditions as inputs, and the solution of the physics-informed neural network as a scalar output for the forward design of the neural network.
• Figure : Fig.2 Partition of Unity Networks Leading to PINN: This figure shows our approach by using the data and the input dimensions as a vector of inputs $(x \text{ and } y)$. Then, the partition of unity network outputs as it learns the $c$ learnable parameters and $\varphi$ functions are used to compute the conductivity using the following equation, which is then inserted into the forward-designed physics-informed neural network.
• Figure : Fig.3 Partition of Unity Networks Architecture: The architecture of the partition of the unity network consists of inputs, hidden layer(s), a linear activation function layer, and a softmax activation function layer, leading to the output functions $\vec{\varphi}$.
• Figure : Fig.4. Partition of Unity Network Physics-Informed Neural Networks: Here, we show a schematic representation of the partition of unity networks and the physics-informed neural networks and how they are connected to create a partition of unity physics-informed neural networks.
• Figure : Fig.5 Scalar field PDE solution: Scalar field solution is computed using the forward design of PINNs. The solution demonstrates four subdomains with equal geometric characteristics and two pairs with equal solutions on different subdomains. Then, the three-dimensional representation of our solution is on the right.