Partition of Unity Physics-Informed Neural Networks (POU-PINNs) for Darcy Flow

Partition of Unity Physics-Informed Neural Networks (POU-PINNs) represent a fusion of classical numerical analysis and modern machine learning, aiming to address the challenges of solving parametrized partial differential equations in heterogeneous domains. Unlike traditional PINNs, which exhibit convergence and accuracy difficulties in the presence of spatial discontinuities, the POU-PINN architecture supports domain decomposition in partition-of-unity theory, facilitating locality- and physics-based approaches. Each subdomain features spatially varying learning parameters and is governed by a physics-based residual loss, promoting the unsupervised discovery of latent physical regimes. This formulation enables scalability to multiscale conductivity problems, as demonstrated in pore mass ablation and ice sheet dynamics. By disentangling spatial structure and physics through flexible subdomain assignments, the framework provides interpretable expert mixtures that seamlessly respect interface conditions and flow continuity. Our work contributes to the growing body of research focused on physics-based unsupervised learning and surrogate probabilistic modeling for high-consequence systems, enabling the integration of scientific constraints directly into learning architectures for predictive simulation under uncertainty. In this work, we present POU-PINNs, a physics solver for partial differential equations that utilizes neural networks, enabling the simulation of discontinuities through the partition of unity method.