  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.
