MgNO: Efficient Parameterization of Linear Operators via Multigrid
Juncai He Xinliang Liu Jinchao Xu
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 $\mathcal O_i(u) =
\sigma\left( \sum_j \mathcal W_{ij} u + \mathcal B_{ij}\right)$. Here,
$\mathcal W_{ij}$ denotes the bounded linear operator connecting $j$-th input
neuron to $i$-th output neuron, and the bias $\mathcal B_{ij}$ 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).