  This paper proposes a meshless deep learning algorithm, enriched
physics-informed neural networks (EPINNs), to solve dynamic
Poisson-Nernst-Planck (PNP) equations with strong coupling and nonlinear
characteristics. The EPINNs takes the traditional physics-informed neural
networks as the foundation framework, and adds the adaptive loss weight to
balance the loss functions, which automatically assigns the weights of losses
by updating the parameters in each iteration based on the maximum likelihood
estimate. The resampling strategy is employed in the EPINNs to accelerate the
convergence of loss function. Meanwhile, the GPU parallel computing technique
is adopted to accelerate the solving process. Four examples are provided to
demonstrate the validity and effectiveness of the proposed method. Numerical
results indicate that the new method has better applicability than traditional
numerical methods in solving such coupled nonlinear systems. More importantly,
the EPINNs is more accurate, stable, and fast than the traditional
physics-informed neural networks. This work provides a simple and
high-performance numerical tool for addressing PNPs with arbitrary boundary
shapes and boundary conditions.
