Work

HMgNO: Hybrid multigrid neural operator with low-order numerical solver for partial differential equations

Year: 2025

Type: article

Abstract: Traditional numerical methods face a trade-off between computational cost and accuracy when solving partial differential equations. Low-order solvers are fast but less accurate, while high-order solvers are accurate but much slower. To address this challenge, we propose a novel framework, the hybrid multigrid neural operator (HMgNO). The HMgNO couples a low-order numerical solver with a multigrid neural operator, and the neural operator is used to correct the low-order numerical solutions to obtain high-order accuracy at each fixed time step size. Thus, the HMgNO achieves accurate solutions while ensuring computational efficiency. Moreover, our framework supports multiple types of low-order numerical solvers, such as finite difference and spectral methods. Experiments on the Navier-Stokes, shallow-water, and diffusion-reaction equations demonstrate that the proposed framework achieves the lowest relative error and smallest spectral bias with few model parameters and fast inference speed.

Source: Neural Networks

DOI: 10.1016/j.neunet.2025.107649

Authors: Yifan Hu, Weimin Zhang, Fukang Yin, Jianping Wu

Institution: National University of Defense Technology

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,110

Funder: National Natural Science Foundation of China