Operator Learning: Algorithms and Analysis
It addresses the need for efficient surrogate models in many-query tasks for researchers in computational science and engineering, but is incremental as it reviews existing progress without introducing new methods.
This review tackles the problem of approximating nonlinear operators in Banach spaces, such as those from PDEs, using neural operators as data-driven surrogate models, highlighting their empirical success in applications but noting incomplete theoretical understanding.
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.