Time-marching neural operator–FE coupling : AI-accelerated physics modeling

Abstract

Numerical solvers for partial differential equations (PDEs) often struggle to balance computational efficiency with accuracy, especially in multiscale and time-dependent systems. Neural operators offer a promising avenue to accelerate simulations, but their practical deployment is hindered by several challenges: they typically require large volumes of training data generated from high-fidelity solvers, tend to accumulate errors over time in dynamical settings, and often exhibit poor generalization in multiphysics scenarios. This work introduces a novel hybrid framework that integrates physics-informed deep operator network (PI-DeepONet) with finite element method (FEM) through domain decomposition and leverages numerical analysis for time marching. The core innovation lies in efficient coupling FEM and DeepONet subdomains via a Schwarz alternating method, expecting to solve complex and nonlinear regions by a pre-trained DeepONet, while the remainder is handled by conventional FEM. To address the challenges of dynamic systems, we embed the Newmark-Beta time-stepping scheme directly into the DeepONet architecture, substantially reducing long-term error propagation. Furthermore, an adaptive subdomain evolution strategy enables the ML-resolved region to expand dynamically, capturing emerging fine-scale features without remeshing. The framework's efficacy has been rigorously validated across a range of solid mechanics problems—spanning static, quasi-static, and dynamic regimes including linear elasticity, hyperelasticity, and elastodynamics—demonstrating accelerated convergence rates (up to 20 % improvement in convergence rates compared to conventional FE coupling approaches) while preserving solution fidelity with error margins consistently below 3 %. Our extensive case studies demonstrate that our proposed hybrid solver: (1) reduces computational costs by eliminating fine mesh requirements, (2) mitigates error accumulation in time-dependent simulations, and (3) enables automatic adaptation to evolving physical phenomena. This work establishes a new paradigm for coupling state-of-the-art physics-based and machine learning solvers in a unified framework—offering a robust, reliable, and scalable pathway for high-fidelity multiscale simulations.