  Operator learning aims to discover properties of an underlying dynamical
system or partial differential equation (PDE) from data. Here, we present a
step-by-step guide to operator learning. We explain the types of problems and
PDEs amenable to operator learning, discuss various neural network
architectures, and explain how to employ numerical PDE solvers effectively. We
also give advice on how to create and manage training data and conduct
optimization. We offer intuition behind the various neural network
architectures employed in operator learning by motivating them from the
point-of-view of numerical linear algebra.
