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).

翻译：本文提出了一种用于算子学习的简洁神经算子架构。通过与传统的全连接神经网络进行类比，我们将神经算子定义如下：非线性算子层中第$i$个神经元的输出由$\mathcal O_i(u) = \sigma\left( \sum_j \mathcal W_{ij} u + \mathcal B_{ij}\right)$定义。其中，$\mathcal W_{ij}$表示连接第$j$个输入神经元与第$i$个输出神经元的有界线性算子，偏置项$\mathcal B_{ij}$采用函数形式而非标量形式。基于其新的通用逼近特性，两个神经元（巴拿赫空间）间有界线性算子的高效参数化至关重要。为此，我们提出MgNO，利用多重网格结构对神经元间的线性算子进行参数化。该方法兼具数学严谨性与实际表达能力。此外，MgNO无需传统神经算子中通常所需的升维与投影算子，并能自然地适应各类边界条件。实验观察表明，相较于其他基于CNN的模型，MgNO具有更优的训练便利性；与谱类神经算子相比，其过拟合倾向显著降低。我们通过在多种偏微分方程（PDE）问题上持续取得最先进性能的实验结果，验证了该方法的高效性与准确性。