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

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.