Improved multi-scale fusion network for solving non-smooth elliptic interface problems with applications

The utilization of deep learning methodologies for addressing partial differential equations (PDEs) has garnered significant attention in recent years. This paper introduces an improved network structure tailored for the discontinuity -capturing, enabling the resolution of interface problem through a unified neural network framework. Employing the probability space filling argument, we show that our model can generate convergent sequences, where the convergence rate depends on the number of sampling points. Several numerical experiments with regular and irregular interfaces are conducted to elucidate the convergence characteristics, thereby validating the theoretical assertions. Furthermore, we apply our approach to effectively solve the size -modified Poisson -Boltzmann test model, utilizing it for predicting electrostatics and the solvation free energies for proteins immersed in ionic solvents, thus showcasing practical applications of our method.