@Article{NMTMA-16-883,
author = {Hou , MuzhouChen , YinghaoCao , ShenChen , Yuntian and Ying , Jinyong},
title = {HRW: Hybrid Residual and Weak Form Loss for Solving Elliptic Interface Problems with Neural Network},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2023},
volume = {16},
number = {4},
pages = {883--913},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2023-0097},
url = {http://global-sci.org/intro/article_detail/nmtma/22115.html}
}