# Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modeling

arXiv: https://arxiv.org/abs/2504.11383 · HTML: https://mlsanity.com/paper/2504.11383/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 62 references that are in the map.
- First posted on arXiv 15 Apr 2025.

## Builds on (most cited first, 17 of 17)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [On the training and generalization of deep operator networks](https://mlsanity.com/paper/2309.01020/) (2023)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [Rethinking materials simulations: Blending direct numerical simulations with neural operators](https://mlsanity.com/paper/2312.05410/) (2023)
- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics](https://mlsanity.com/paper/2207.09908/) (2022)
- [Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating method](https://mlsanity.com/paper/2311.00224/) (2023)
- [Koopman operator for time-dependent reliability analysis](https://mlsanity.com/paper/2203.02658/) (2022)
- [Causality-Respecting Adaptive Refinement for PINNs: Enabling Precise Interface Evolution in Phase Field Modeling](https://mlsanity.com/paper/2410.20212/) (2024)

## Built on by (most cited first, 8 of 8)

- [Enabling Local Neural Operators to perform Equation-Free System-Level Analysis](https://mlsanity.com/paper/2505.02308/) (2025)
- [Hybrid coupling with operator inference and the overlapping Schwarz alternating method](https://mlsanity.com/paper/2511.20687/) (2025)
- [Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators](https://mlsanity.com/paper/2509.06154/) (2025)
- [Multimodal Neural Operators for Real-Time Biomechanical Modelling of Traumatic Brain Injury](https://mlsanity.com/paper/2510.03248/) (2025)
- [Dmsh: A Multi-Agent Reinforcement Learning Framework for All-Quad Mesh Generation](https://mlsanity.com/paper/2606.10601/) (2026)
- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)
- [A Numerical Method for Coupling Parameterized Physics-Informed Neural Networks and FDM for Advanced Thermal-Hydraulic System Simulation](https://mlsanity.com/paper/2604.02663/) (2026)
- [Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees](https://mlsanity.com/paper/2608.02036/) (2026)

## Abstract (arXiv)

Numerical solvers for PDEs often struggle to balance computational cost with accuracy, especially in multiscale and time-dependent systems. Neural operators offer a promising way 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 with FEM through domain decomposition and leverages numerical analysis for time marching. Our innovation lies in efficient coupling FE and DeepONet subdomains via a Schwarz method, expecting to solve complex and nonlinear regions by a pretrained DeepONet, while the remainder is handled by conventional FE. To address the challenges of dynamic systems, we embed a time stepping scheme directly into the DeepONet, substantially reducing long-term error propagation. Furthermore, an adaptive subdomain evolution strategy enables the ML-resolved region to expand dynamically, capturing fine-scale features without remeshing. Our framework shows 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 study shows 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.