Data-driven Approaches And Operator Learning For Solving PDE-related Problems

Solving partial differential equations (PDEs) and PDE-based model reduction are challenging problems, particularly when PDEs have multiscale features. The data-driven approach has become an excellent option for some scientific computing problems. It becomes even more effective for some engineering applications with available data. There are various data-driven treatments for PDE-related problems. Many of them can be implemented in the operator learning framework as the underlying mathematical computation problems construct the operator. I will focus on and discuss operator learning. In particular, I will introduce a new framework: basis enhanced learning (Bel). Bel does not require a specific discretization of functions and achieves great prediction accuracy. Universal approximation theory and some applications, including some newly proposed engineering applications, will be discussed.