MgNO: Efficient Parameterization of Linear Operators via Multigrid

TL;DR

MgNO introduces a multigrid-based neural operator architecture that efficiently parameterizes linear operators, achieving superior training ease, reduced overfitting, and state-of-the-art PDE solving performance.

Contribution

The paper presents MgNO, a novel neural operator using multigrid structures for efficient linear operator parameterization, eliminating the need for lifting/projection and handling diverse boundary conditions.

Findings

MgNO outperforms existing models in training ease.

MgNO demonstrates reduced overfitting compared to spectral neural operators.

MgNO achieves state-of-the-art accuracy on various PDE problems.

Abstract

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a nonlinear operator layer is defined by $O_i(u) = \sigma\left( \sum_j W_{ij} u + B_{ij}\right)$. Here, $ W_{ij}$ denotes the bounded linear operator connecting $j$-th input neuron to $i$-th output neuron, and the bias $ B_{ij}$ takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).