  Operator learning refers to the application of ideas from machine learning to
approximate (typically nonlinear) operators mapping between Banach spaces of
functions. Such operators often arise from physical models expressed in terms
of partial differential equations (PDEs). In this context, such approximate
operators hold great potential as efficient surrogate models to complement
traditional numerical methods in many-query tasks. Being data-driven, they also
enable model discovery when a mathematical description in terms of a PDE is not
available. This review focuses primarily on neural operators, built on the
success of deep neural networks in the approximation of functions defined on
finite dimensional Euclidean spaces. Empirically, neural operators have shown
success in a variety of applications, but our theoretical understanding remains
incomplete. This review article summarizes recent progress and the current
state of our theoretical understanding of neural operators, focusing on an
approximation theoretic point of view.
