Title: Neural-network-based methods for solving free boundary problems

 Abstract: Free boundary problems (the time-dependent versions are also often known as moving boundary problems) deal with systems of partial differential equations (PDEs) where the domain boundary is apriori unknown. Due to this special characteristic, it is challenging to solve free boundary problem numerically, and most studies in this field lack convergence proofs for the numerical methods. In this talk, I will present novel approaches based on neural networks to study two types of free boundary problems: 1) the classical obstacle problem, and 2) a modified Hele-Shaw problem. For each method we proposed, we established the convergence of the scheme and theoretically derived the convergence rate with the number of neurons. Several simulation examples are used to demonstrate the feasibility and capability of the proposed methods.