  The classical development of neural networks has primarily focused on
learning mappings between finite dimensional Euclidean spaces or finite sets.
We propose a generalization of neural networks to learn operators, termed
neural operators, that map between infinite dimensional function spaces. We
formulate the neural operator as a composition of linear integral operators and
nonlinear activation functions. We prove a universal approximation theorem for
our proposed neural operator, showing that it can approximate any given
nonlinear continuous operator. The proposed neural operators are also
discretization-invariant, i.e., they share the same model parameters among
different discretization of the underlying function spaces. Furthermore, we
introduce four classes of efficient parameterization, viz., graph neural
operators, multi-pole graph neural operators, low-rank neural operators, and
Fourier neural operators. An important application for neural operators is
learning surrogate maps for the solution operators of partial differential
equations (PDEs). We consider standard PDEs such as the Burgers, Darcy
subsurface flow, and the Navier-Stokes equations, and show that the proposed
neural operators have superior performance compared to existing machine
learning based methodologies, while being several orders of magnitude faster
than conventional PDE solvers.
