Enriched physics-informed neural networks for dynamic Poisson-Nernst-Planck systems

This paper proposes a meshless deep learning algorithm, called enriched physics-informed neural networks (EPINNs), to solve dynamic Poisson-Nernst-Planck (PNP) equations characterized by strong coupling and nonlinear behaviors. EPINNs build upon traditional physics-informed neural networks (PINNs) by incorporating an adaptive loss weight mechanism, which automatically adjusts the weights of the loss functions during training, based on maximum likelihood estimation, to achieve balanced optimization. Additionally, a resampling strategy is introduced to accelerate the convergence of the loss function. Four numerical examples are presented to demonstrate the validity and effectiveness of the proposed method. The results show that EPINNs have better applicability than traditional numerical methods in solving such coupled nonlinear systems. Furthermore, EPINNs outperform traditional PINNs in terms of accuracy, stability, and speed. This work provides a robust and efficient numerical tool for solving PNP equations with arbitrary boundary shapes and conditions.