Accelerating Multiscale Modeling with Hybrid Solvers: Coupling FEM and Neural Operators with Domain Decomposition

Abstract: Numerical solvers for PDEs face challenges in balancing computational cost and accuracy, particularly for multiscale and dynamical systems. Neural operators (NOs) can significantly speed up simulations; however, they face challenges such as error accumulation for dynamical systems and limited generalization in multiphysics problems. This work introduces a novel hybrid framework that integrates PI-NO with finite element method (FE) through domain decomposition and leverages numerical analysis for time marching. The core innovation lies in efficient coupling FE and NO subdomains via a Schwarz alternating method: regions with complex, nonlinear, or high-gradient behavior are resolved using a pretrained NO, while the remainder is handled by conventional FE. To address the challenges of dynamic systems, we embed a time-stepping scheme directly into the NO architecture, substantially reducing long-term error propagation. Also, an adaptive subdomain evolution strategy enables the ML resolved region to expand dynamically, capturing emerging fine scale features without remeshing. The framework efficacy has been validated across a range of problems, spanning static, quasi-static, and dynamic regimes (e.g., linear elasticity, hyperelasticity, and elastodynamics), demonstrating accelerated convergence (up to 20% improvement in convergence compared to conventional FE coupling) while preserving solution fidelity with error margins consistently below 1%. Our study shows that our hybrid solver: (1) maintains solution continuity across subdomain interfaces, (2) reduces computational costs by eliminating fine mesh requirements, (3) mitigates error accumulation in time dependent simulations, and (4) enables automatic adaptation to evolving physical phenomena. This work bridges the gap between numerical methods and AI-driven surrogates, offering a scalable pathway for high-fidelity multiscale simulations.