MgFNO: Multi-grid Architecture Fourier Neural Operator for Parametric Partial Differential Equations

TL;DR

MgFNO introduces a multigrid hierarchical architecture for Fourier neural operators, significantly improving training speed and accuracy in solving complex PDEs with high-frequency dynamics, and enabling zero-shot super-resolution.

Contribution

The paper presents a novel multigrid Fourier neural operator architecture with a decoupled training strategy, enhancing efficiency and accuracy over traditional FNOs for PDE solutions.

Findings

Achieves over 89% error reduction on Burgers' equation.

Supports zero-shot super-resolution for high-resolution PDE predictions.

Demonstrates superior performance on Darcy flow and Navier-Stokes equations.

Abstract

Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of partial differential equations (PDEs). This paper proposes a multigrid Fourier neural operator (MgFNO) that accelerates the training of traditional Fourier neural operators through a novel three-level hierarchical architecture. The key innovation of MgFNO lies in its decoupled training strategy employing three distinct networks at different resolution levels: a coarse-level network first learns low-resolution approximations, an intermediate network refines the solution, and a fine-level network achieves high-resolution accuracy. By combining the frequency principle of deep neural networks with multigrid methodology, MgFNO effectively bridges the complementary learning patterns of neural networks (low-to-high frequency) and multigrid methods (high-to-low frequency error reduction).Experimental results demonstrate that MgFNO achieves relative errors of 0.17%, 0.28%, and 0.22% on the Burgers' equation, Darcy flow, and Navier-Stokes equations, respectively, representing reductions of 89%, 71%, and 83% compared to the conventional FNO. Furthermore, MgFNO supports zero-shot super-resolution prediction, enabling direct application to high-resolution scenarios after training on coarse grids. This study establishes an efficient and high-accuracy new paradigm for solving complex PDEs dominated by high-frequency dynamics. Code and data used are available on https://github.com/guozihao-hub/MgFNO/tree/master.