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

TL;DR

MgFNO addresses the bottleneck of training Fourier Neural Operators for parametric PDEs by introducing a three-level V-cycle multigrid framework that trains on coarse grids to capture low-frequency content and progressively learns residual high-frequency components on finer grids. The method preserves FNO's resolution-invariance while rebalancing the learning of frequency components, enabling zero-shot super-resolution and substantial accuracy improvements on Burgers, Darcy, and Navier–Stokes problems. Empirical results show relative errors as low as 0.17%, 0.28%, and 0.22% across these tests, with notably faster convergence than standard FNO. The work offers a practical paradigm for solving complex PDEs with dominant high-frequency dynamics and suggests avenues for extending the theoretical understanding of the F-principle in operator learning and for refining grid-conversion criteria.

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.

Figures (15)

• Figure 1: : FNO introduces a trade-off between the speed of computation and the accuracy of fitting with different "modes" according to the parameter settings in FNO1 to solve 1-d Burger's problem.
• Figure 2: : Distribution of the convergence factor $\mu_{loc}$.
• Figure 3: ： FNO network: The input PDE parameter $a$ is passed from $P$ to $v_0(x)$, then it undergoes iteration in the Fourier space to obtain $V_T(x)$, and finally it goes through $Q$ to reach the spatial dimension of the solution $u$.
• Figure 4: : Training process of DNNs in the spatial domain.
• Figure 5: : Training process of DNNs in the frequency domain. Amplitude $|\hat{y}_k|$ vs. frequency.