Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast, convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically, inspired by stencil methods, we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and, hence, the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance, reducing the relative L2-error by 34-72% in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.

翻译：神经算子学习函数空间之间的映射，这对于学习偏微分方程的解算子及其他科学建模应用具有实际意义。其中，傅里叶神经算子（FNO）是一种流行的架构，其在傅里叶空间执行全局卷积运算。然而，此类全局操作往往容易导致过度平滑，且可能无法捕捉局部细节。相比之下，卷积神经网络（CNN）能够捕捉局部特征，但仅限于单一分辨率下的训练与推理。本文提出了一种原理性的算子学习方法，通过使用局部支撑核学习微分算子与积分算子，能够在两种框架下捕捉局部特征。具体而言，受模板方法启发，我们证明了通过对CNN核值进行适当缩放即可获得微分算子。为获得局部积分算子，我们基于离散-连续卷积为核函数构建了合适的基表示。这两种方法均保持了算子学习的特性，因而具备任意分辨率下的预测能力。将所提出的层添加到FNO中可显著提升其性能，在我们包含二维湍流Navier-Stokes方程和球面浅水方程的实验中，相对L2误差降低了34-72%。