Operator Learning: A Statistical Perspective
This is an incremental review article that formalizes and synthesizes existing operator learning methods for scientific computing applications like PDE modeling.
This paper formalizes operator learning as a function-to-function regression problem for approximating mappings between infinite-dimensional function spaces, primarily for developing surrogate models of PDE solution operators and black-box simulators from experimental data, and reviews recent developments while discussing strategies for incorporating physical constraints and future directions.
Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the solution operators of partial differential equations (PDEs). These methods can also be used to develop black-box simulators to model system behavior from experimental data, even without a known mathematical model. In this article, we begin by formalizing operator learning as a function-to-function regression problem and review some recent developments in the field. We also discuss PDE-specific operator learning, outlining strategies for incorporating physical and mathematical constraints into architecture design and training processes. Finally, we end by highlighting key future directions such as active data collection and the development of rigorous uncertainty quantification frameworks.