  Neural operators have recently become popular tools for designing solution
maps between function spaces in the form of neural networks. Differently from
classical scientific machine learning approaches that learn parameters of a
known partial differential equation (PDE) for a single instance of the input
parameters at a fixed resolution, neural operators approximate the solution map
of a family of PDEs. Despite their success, the uses of neural operators are so
far restricted to relatively shallow neural networks and confined to learning
hidden governing laws. In this work, we propose a novel nonlocal neural
operator, which we refer to as nonlocal kernel network (NKN), that is
resolution independent, characterized by deep neural networks, and capable of
handling a variety of tasks such as learning governing equations and
classifying images. Our NKN stems from the interpretation of the neural network
as a discrete nonlocal diffusion reaction equation that, in the limit of
infinite layers, is equivalent to a parabolic nonlocal equation, whose
stability is analyzed via nonlocal vector calculus. The resemblance with
integral forms of neural operators allows NKNs to capture long-range
dependencies in the feature space, while the continuous treatment of
node-to-node interactions makes NKNs resolution independent. The resemblance
with neural ODEs, reinterpreted in a nonlocal sense, and the stable network
dynamics between layers allow for generalization of NKN's optimal parameters
from shallow to deep networks. This fact enables the use of shallow-to-deep
initialization techniques. Our tests show that NKNs outperform baseline methods
in both learning governing equations and image classification tasks and
generalize well to different resolutions and depths.
