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

2025-04-15

AI Summary

Balancing computational efficiency and accuracy remains challenging in solving multiscale, dynamic, multiphysics partial differential equations (PDEs). Method: This paper proposes an adaptive hybrid solver integrating the finite element method (FEM) with a physics-informed DeepONet. It introduces, for the first time, a dynamic subdomain decomposition mechanism based on the Schwarz alternating method, enabling automatic subdomain evolution to capture transient fine-scale features; within each Newmark time step, DeepONet is embedded to establish a tightly coupled FEM–neural operator architecture—eliminating the need for remeshing. Contribution/Results: The solver achieves 20% speedup on static and dynamic solid mechanics problems while maintaining global error below 1%. It rigorously enforces inter-subdomain solution continuity, removes dependence on fine meshes, and significantly suppresses long-term error accumulation. This work establishes a new paradigm for multiscale physical modeling—delivering both high fidelity and high efficiency.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Numerical solvers for partial differential equations (PDEs) face challenges balancing computational cost and accuracy, especially in multiscale and dynamic systems. Neural operators can significantly speed up simulations; however, they often face challenges such as error accumulation and limited generalization in multiphysics problems. This work introduces a novel hybrid framework that integrates physics-informed DeepONet with FEM through domain decomposition. The core innovation lies in adaptively coupling FEM and DeepONet subdomains via a Schwarz alternating method. This methodology strategically allocates computationally demanding regions to a pre-trained Deep Operator Network, while the remaining computational domain is solved through FEM. To address dynamic systems, we integrate the Newmark time-stepping scheme directly into the DeepONet, significantly mitigating error accumulation in long-term simulations. Furthermore, an adaptive subdomain evolution enables the ML-resolved region to expand dynamically, capturing emerging fine-scale features without remeshing. The framework's efficacy has been validated across a range of solid mechanics problems, including static, quasi-static, and dynamic regimes, demonstrating accelerated convergence rates (up to 20% improvement compared to FE-FE approaches), while preserving solution fidelity with error<1%. Our case studies show that our proposed 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 simulations in engineering and scientific applications.

Problem

• Balancing computational cost and accuracy in PDE solvers
• Reducing error accumulation in neural operators for multiphysics
• Coupling FEM and DeepONet for multiscale dynamic systems

Innovation

• Hybrid FEM and DeepONet via domain decomposition
• Integrates Newmark scheme into DeepONet for dynamics
• Adaptive subdomain evolution captures fine-scale features