Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains
Ziqi Liu, Wei Cai, Zhi-Qin John Xu
TL;DR
The paper tackles solving Poisson-Boltzmann and Poisson equations in complex and singular domains with high-frequency solution content, where standard DNNs struggle. It introduces MscaleDNN, a mesh-less approach that uses radial frequency scaling and compact activation functions to decompose learning across multiple frequency bands, paired with Ritz variational training for PB problems. Two architectures (MscaleDNN-1 and MscaleDNN-2) and a Ritz-based loss are developed, demonstrating faster convergence and lower errors than conventional DNNs across a wide range of test problems, including variable coefficients, ring and perforated domains, and geometric or source singularities. The results indicate significant potential for scalable, mesh-free PDE solvers in computational physics and chemistry, with future work exploring wavelet-inspired activations and very high-dimensional applications.
Abstract
In this paper, we propose multi-scale deep neural networks (MscaleDNNs) using the idea of radial scaling in frequency domain and activation functions with compact support. The radial scaling converts the problem of approximation of high frequency contents of PDEs' solutions to a problem of learning about lower frequency functions, and the compact support activation functions facilitate the separation of frequency contents of the target function to be approximated by corresponding DNNs. As a result, the MscaleDNNs achieve fast uniform convergence over multiple scales. The proposed MscaleDNNs are shown to be superior to traditional fully connected DNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.