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

• URL: http://arxiv.org/abs/2504.11383v3
• Date: Wed, 14 May 2025 15:14:24 GMT
• Title: Accelerating Multiscale Modeling with Hybrid Solvers: Coupling FEM and Neural Operators with Domain Decomposition
• Authors: Wei Wang, Maryam Hakimzadeh, Haihui Ruan, Somdatta Goswami,
• Abstract summary: This work introduces a novel hybrid framework that integrates PI-NO with finite element method (FE) through domain decomposition.<n>The framework efficacy has been validated across a range of problems, spanning static, quasi-static, and dynamic regimes.<n>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.
• Score: 3.0635300721402228
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations [4.7903561901859355]
  We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step.<n>Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase.<n>Across three representative PDE systems, PhysicsCorrect reduces prediction errors by up to 100x while adding negligible inference time.
  arXiv  Detail & Related papers  (2025-07-03T01:22:57Z)
• Implicit Neural Differential Model for Spatiotemporal Dynamics [5.1854032131971195]
  We introduce Im-PiNDiff, a novel implicit physics-integrated neural differentiable solver for stabletemporal dynamics.<n>Inspired by deep equilibrium models, Im-PiNDiff advances the state using implicit fixed-point layers, enabling robust long-term simulation.<n>Im-PiNDiff achieves superior predictive performance, enhanced numerical stability, and substantial reductions in memory and cost.
  arXiv  Detail & Related papers  (2025-04-03T04:07:18Z)
• Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems [49.819436680336786]
  Gaussian process state-space models (GPSSMs) have emerged as a powerful framework for modeling dynamical systems.<n>We propose an efficient transformed Gaussian process state-space model (ETGPSSM) to address these limitations.<n>Our approach leverages a single shared Gaussian process (GP) combined with normalizing flows and Bayesian neural networks, enabling efficient modeling of complex, high-dimensional state transitions.
  arXiv  Detail & Related papers  (2025-03-24T03:19:45Z)
• Multi-Agent Path Finding in Continuous Spaces with Projected Diffusion Models [57.45019514036948]
  Multi-Agent Path Finding (MAPF) is a fundamental problem in robotics.<n>This work proposes a novel approach that integrates constrained optimization with diffusion models for MAPF in continuous spaces.
  arXiv  Detail & Related papers  (2024-12-23T21:27:19Z)
• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs)<n>We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations.<n>We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks [0.24578723416255746]
  We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. A distinct advantage our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations.
  arXiv  Detail & Related papers  (2024-09-20T16:48:55Z)
• A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equations [2.7755345520127936]
  We propose a domain-decomposition-based deep learning (DL) framework, named CoMLSim, for accurately modeling unsteady and nonlinear partial differential equations (PDEs)<n>The framework consists of two key components: (a) a convolutional neural network (CNN)-based autoencoder architecture and (b) an autoregressive model composed of fully connected layers.
  arXiv  Detail & Related papers  (2024-08-26T17:50:47Z)
• Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics [1.7132914341329852]
  We present the neural-integrated meshfree (NIM) method, a differentiable programming-based hybrid meshfree approach within the field of computational mechanics. NIM seamlessly integrates traditional physics-based meshfree discretization techniques with deep learning architectures. Under the NIM framework, we propose two truly meshfree solvers: the strong form-based NIM (S-NIM) and the local variational form-based NIM (V-NIM)
  arXiv  Detail & Related papers  (2023-11-21T17:57:12Z)
• A Multi-Head Ensemble Multi-Task Learning Approach for Dynamical Computation Offloading [62.34538208323411]
  We propose a multi-head ensemble multi-task learning (MEMTL) approach with a shared backbone and multiple prediction heads (PHs) MEMTL outperforms benchmark methods in both the inference accuracy and mean square error without requiring additional training data.
  arXiv  Detail & Related papers  (2023-09-02T11:01:16Z)
• Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction [49.66486092259376]
  The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the Langevin dynamics that incorporates a distribution-dependent drift. Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures. We provide a framework to prove a uniform-in-time propagation of chaos for MFLD that takes into account the errors due to finite-particle approximation, time-discretization, and gradient approximation.
  arXiv  Detail & Related papers  (2023-06-12T16:28:11Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Multi-fidelity Hierarchical Neural Processes [79.0284780825048]
  Multi-fidelity surrogate modeling reduces the computational cost by fusing different simulation outputs. We propose Multi-fidelity Hierarchical Neural Processes (MF-HNP), a unified neural latent variable model for multi-fidelity surrogate modeling. We evaluate MF-HNP on epidemiology and climate modeling tasks, achieving competitive performance in terms of accuracy and uncertainty estimation.
  arXiv  Detail & Related papers  (2022-06-10T04:54:13Z)
• Multilayer Perceptron Based Stress Evolution Analysis under DC Current Stressing for Multi-segment Wires [8.115870370527324]
  Electromigration (EM) is one of the major concerns in the reliability analysis of very large scale integration (VLSI) systems. Traditional methods are often not sufficiently accurate, leading to undesirable over-design especially in advanced technology nodes. We propose an approach using multilayer perceptrons (MLP) to compute stress evolution in the interconnect trees during the void nucleation phase.
  arXiv  Detail & Related papers  (2022-05-17T07:38:20Z)
• Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems [4.280301926296439]
  In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown fine-scale dynamics. We present various benchmarks to assess accuracy and speedup, and in particular we develop a coupling algorithm for a time-dependent problem.
  arXiv  Detail & Related papers  (2022-02-25T20:46:08Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Physics and Equality Constrained Artificial Neural Networks: Application to Partial Differential Equations [1.370633147306388]
  Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE) Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach. We propose a versatile framework that can tackle both inverse and forward problems.
  arXiv  Detail & Related papers  (2021-09-30T05:55:35Z)
• Efficient Model-Based Multi-Agent Mean-Field Reinforcement Learning [89.31889875864599]
  We propose an efficient model-based reinforcement learning algorithm for learning in multi-agent systems. Our main theoretical contributions are the first general regret bounds for model-based reinforcement learning for MFC. We provide a practical parametrization of the core optimization problem.
  arXiv  Detail & Related papers  (2021-07-08T18:01:02Z)
• Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction [79.81193813215872]
  We develop a hybrid (graph) neural network that combines a traditional graph convolutional network with an embedded differentiable fluid dynamics simulator inside the network itself. We show that we can both generalize well to new situations and benefit from the substantial speedup of neural network CFD predictions.
  arXiv  Detail & Related papers  (2020-07-08T21:23:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.