HRW: Hybrid Residual and Weak Form Loss for Solving Elliptic Interface Problems with Neural Network

Deep learning techniques for solving elliptic interface problems have gained significant attentions. In this paper, we introduce a hybrid residual and weak form (HRW) loss aimed at mitigating the challenge of model training. HRW utilizes the functions residual loss and Ritz method in an adversary-system, which enhances the probability of jumping out of the local optimum even when the loss landscape comprises multiple soft constraints (regularization terms), thus improving model's capability and robustness. For the problem with interface conditions, unlike existing methods that use the domain decomposition, we design a Pre-activated ResNet of ResNet (PRoR) network structure employing a single network to feed both coordinates and corresponding subdomain indicators, thus reduces the number of parameters. The effectiveness and improvements of the PRoR with HRW are verified on two-dimensional interface problems with regular or irregular interfaces. We then apply the PRoR with HRW to solve the size-modified Poisson-Boltzmann equation, an improved dielectric continuum model for predicting the electrostatic potentials in an ionic solvent by considering the steric effects. Our findings demonstrate that the PRoR with HRW accurately approximates solvation free-energies of three proteins with irregular interfaces, showing the competitive results compared to the ones obtained using the finite element method.