ABSTRACT

Recently, there have been extensive studies on neural network-based method for solving partial differential equations (PDEs). Among them, physics-informed neural network (PINN) is one of the prominent approaches, in which the governing equations are explicitly included in in loss functionals. The main reason behind such success is the universal approximation property. However, conventional PINNs encounter difficulties when the solution have discontinuity across some interface. In this talk, we present a remedy to address this issue. We introduce an axis-extension technique to represent solutions in continuous form in higher dimension. Based on this representation, neural network function now can enjoy universal approximation property. One advantage of such approach is the availability of convergence analysis, since the representation of solution with extended axis is smooth. In this talk, we provide two examples. We solve Poisson-Boltzmann equation and heat equation with implicit jump conditions. We describe how we derived error estimates. Also, we provide some numerical experiments.