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 on turbulent 2D Navier-Stokes fluid flow 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解算子及其他科学建模应用具有实用价值。其中，傅里叶神经算子（FNO）是一种在傅里叶空间执行全局卷积的流行架构。然而，此类全局操作常易导致过度平滑，且难以捕捉局部细节。相比之下，卷积神经网络（CNN）虽能捕捉局部特征，却受限于在固定分辨率下进行训练与推理。本文提出一种原则性的算子学习方法，通过学习微分算子及具有局部支撑核的积分算子，在两类框架下均可捕捉局部特征。具体而言，受模板方法启发，我们证明在对CNN核值进行适当缩放后即可获得微分算子。为获得局部积分算子，我们基于离散-连续卷积采用合适的核基表示。这两种方法均保持了算子学习的特性，因此具备在任意分辨率下进行预测的能力。将我们的层模块融入FNO将显著提升其性能：在基于二维湍流Navier-Stokes流体流动与球面浅水方程的实验中，相对L2误差降低了34%-72%。