Efficient and scalable galerkin neural networks for multiphysics problems via domain decomposition

Abstract In this work, we develop an efficient machine-learning–assisted framework for the approximation of partial differential equations arising in multiphysics science and engineering. Building on the Galerkin Neural Network (GNN) formulation for symmetric, positive-definite problems and leveraging the well-established efficiency of domain decomposition methods (DDM), we introduce a coupled GNN–DDM strategy designed to enhance scalability and computational performance. The proposed framework integrates data-driven trial spaces with physics-based subdomain solvers, enabling scalable training and solution procedures for large-scale multiphysics systems. We establish convergence of the coupled method using previous results, demonstrate its performance on a suite of benchmark multiphysics problems, and provide a flexible computational infrastructure built on modern Python libraries for reproducible and extensible implementation. Results show that the GNN–DDM approach achieves accurate, scalable, and efficient solutions, highlighting its potential as a practical tool for next-generation multiphysics simulation.