| Contents | Partial differential equations (PDEs) with interfaces arise in many scientific and engineering applications, including multiphase flows, fluid–structure interactions, and moving boundary problems. In this talk, I will first introduce some interesting PDEs with interfaces and talk about their numerical difficulties from classical numerical point of view. As known, the presence of interfaces introduces discontinuities in solutions or their derivatives, in which the grid-based method must carefully discretize the equations near the interface in order to keep certain accuracy. This task becomes even challenging when the interface is moving. Based on mesh-free advantage, neural network method provides an alternative to solve such kind of problems. Here, I will introduce a feature-capturing neural network methodology to solve steady elliptic interface problems and unsteady Stefan problems. Our numerical tests show competitive performance in terms of accuracy while the implementation is much easier. Some future research directions will also be discussed.
|
