Colloquium : Learning Operators

Abstract

Operator Learning is a new field within machine learning where the aim is to learn operators, such as the solution operators corresponding to forward or inverse problems of PDEs, from data. A key distinction with traditional deep learning lies in the fact that the inputs and outputs of operators are infinite-dimensional. In this talk, we will introduce the concept of representation equivalent neural operators or ReNO which is based on a tight correspondence between the underlying continuous operator and its (multiple) discretizations. Existing operator learning frameworks are analyzed vis a vis this notion. A new model, Convolutional Neural Operators (CNOs) are also introduced which instantiate ReNOs within the class of convolutional neural networks. Extensive numerical experiments showing that CNO is state of the art for operator learning is demonstrated on a wide variety of PDE benchmarks. If time permits, we will discuss applications of operator learning to learn statistical solutions of chaotic PDEs using score based diffusion models and the approximation of inverse problems for PDEs.