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.

翻译：本文提出了一种用于求解改进的一维Poisson-Nernst-Planck离子通道（PNPic）模型的深度学习方法，称为PNPic深度学习求解器。该方法创新性地将一种新型局部神经网络方案与高效的PNPic有限元求解器相结合。由于神经网络方案的输入数据仅涉及有限元求解器可快速生成的局部粗网格解小块，因此PNPic深度学习求解器的训练速度远快于任何传统的全局神经网络求解器。经过充分训练后，该求解器能够以远高于低成本粗网格解的精度输出预测的PNPic解，并能反映参数、离子通道子区域、界面及边界值等不同扰动情况。因此，PNPic深度学习求解器可为一系列PNPic模型生成高精度的数值解。作为初步研究，我们分别通过扰动PNPic模型的一个和两个参数进行了两类数值测试，并以模型中的若干扰动界面位置作为训练样本进行了测试。这些测试表明，PNPic深度学习求解器能够生成高精度的PNPic数值解。