A Mathematical Guide to Operator Learning
It offers a tutorial for researchers in scientific computing and machine learning, but is incremental as it synthesizes existing knowledge without introducing new methods.
The paper provides a comprehensive guide to operator learning, explaining problem types, neural network architectures, and integration with numerical PDE solvers, but does not report specific experimental results or numbers.
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.