XPINN

XPINN stands for eXtended Physics-Informed Neural Network. It is a domain-decomposition extension of Physics-Informed Neural Networks (PINNs) designed to solve partial differential equations by embedding physical laws into neural network training. In XPINN, the computational domain is partitioned into multiple subdomains. A separate neural network is assigned to each subdomain, allowing local approximations of the solution that can be trained independently or in parallel.

Interface conditions are imposed at subdomain boundaries to ensure global consistency. These conditions typically express continuity of the solution and conservation of flux and are enforced via penalty terms, Lagrange multipliers, or multiphysics constraints in the loss function. The XPINN training thus couples the subnets through these interface terms, enabling information exchange across subdomains while preserving some degree of independence to improve scalability.

XPINN extends PINNs to handle heterogeneous materials, complex geometries, and multi-physics problems where a single global network may struggle with sharp gradients or stiffness. It supports parallel computation and can adapt subdomain granularity to problem features, potentially improving convergence and accuracy for high-dimensional or time-dependent PDEs.

Typical applications include fluid dynamics, wave propagation, diffusion-reaction systems, and coupled multi-physics problems. XPINN relies on automatic differentiation to compute residuals of the governing PDEs within each subdomain and on collocation points or other sampling schemes to train the networks.

As with PINNs, XPINN benefits from physics-informed loss terms but requires careful tuning of interface penalties and subdomain partitioning to balance accuracy and computational efficiency.