  In this paper, a deep learning method for solving an improved one-dimensional
Poisson-Nernst-Planck ion channel (PNPic) model, called the PNPic deep learning
solver, is presented. In particular, it combines a novel local neural network
scheme with an effective PNPic finite element solver. Since the input data of
the neural network scheme only involves a small local patch of coarse grid
solutions, which the finite element solver can quickly produce, the PNPic deep
learning solver can be trained much faster than any corresponding conventional
global neural network solvers. After properly trained, it can output a
predicted PNPic solution in a much higher degree of accuracy than the low cost
coarse grid solutions and can reflect different perturbation cases on the
parameters, ion channel subregions, and interface and boundary values, etc.
Consequently, the PNPic deep learning solver can generate a numerical solution
with high accuracy for a family of PNPic models. As an initial study, two types
of numerical tests were done by perturbing one and two parameters of the PNPic
model, respectively, as well as the tests done by using a few perturbed
interface positions of the model as training samples. These tests demonstrate
that the PNPic deep learning solver can generate highly accurate PNPic
numerical solutions.
