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

Area of Science:

• Computational mathematics
• Scientific machine learning
• Numerical analysis

Background:

• Traditional numerical methods for partial differential equations (PDEs) present a trade-off between computational cost and accuracy.
• Low-order solvers are computationally efficient but lack accuracy, while high-order solvers are accurate but computationally expensive.

Purpose of the Study:

• To develop a novel framework that overcomes the accuracy-efficiency trade-off in solving PDEs.
• To enhance the accuracy of low-order numerical solvers using neural operators.

Main Methods:

• The proposed framework, hybrid multigrid neural operator (HMgNO), couples a low-order numerical solver with a multigrid neural operator.
• The neural operator corrects low-order solutions to achieve high-order accuracy at fixed time step sizes.
• The framework is versatile, supporting various low-order solvers like finite difference and spectral methods.

Main Results:

• Experiments on Navier-Stokes, shallow-water, and diffusion-reaction equations demonstrate superior performance.
• The HMgNO framework achieved the lowest relative error and smallest spectral bias.
• The method requires few model parameters and offers fast inference speeds.

Conclusions:

• The HMgNO framework effectively achieves high-order accuracy and computational efficiency for solving PDEs.
• This hybrid approach offers a promising solution for complex scientific simulations.
• The HMgNO framework presents a significant advancement in numerical methods for PDEs.