The data-driven approach has become an excellent option for some scientific computing problems. 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 discuss some theoretical extensions on the classical structure and introduce a new framework: basis enhanced learning (Bel). Bel does not require a specific discretization of the input and output functions and achieves great prediction accuracy. Universal approximation theory and some applications, including some newly proposed engineering applications, will be discussed.