A PNP ion channel deep learning solver with local neural network and finite element input data

TL;DR

This work addresses the computational challenge of solving parameter-sensitive 1D PNPic models for ion channels. It introduces a solver that couples a new 1D finite element solver with a local neural network (NNLCI) to transform low-cost coarse-grid FE solutions into high-fidelity predictions across parameter perturbations and variable interfaces. The approach yields accurate, perturbation-tolerant solutions with significantly reduced training data requirements compared to global networks, and demonstrates robustness to interface shifts, pointing to scalable extension toward 3D ion-channel models. This has practical impact for rapid, parametric ion-channel studies and future multi-dimensional PDE simulations in biophysics.

Abstract

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.

Figures (4)

• Figure 1: A cross section of the K$^+$ ion channel model used in gardner2004electrodiffusionchao2023integral. Here the channel pore domain consists of the interior and exterior conical baths and the cylindrical channel while the channel is further split into the buffer, nonpolar, central cavity, and selectivity filter subdomains to characterize its biochemical properties.
• Figure 2: Numerical solution accuracy improvements by our PNPic deep learning solver in terms of the reduction factor $\tau_i$ defined in \ref{['improvement']}. Here 18 prediction test cases were done by our PNPic deep learning solver for the five parameters $D_{c}, D_{f}$, $\epsilon_c$, $\epsilon_f$, and $Q$, respectively.
• Figure 3: Comparison of the predicted solution functions $\phi_p$, $c_{1,p}$, and $c_{2,p}$ of the PNPic model by our PNPic deep learning solver with the reference solution functions $\phi$, $c_{1}$, and $c_{2}$ generated by our PNPic finite element software package on the fine grid mesh with mesh size $h_{ref}=1/640$ for the test case with $-7\%$ perturbation of the permittivity constant $\epsilon_c$ in the cavity region.
• Figure 4: Comparison of the error functions $|\phi(x) - \phi_p(x)|$ and $|c_i(x) - c_{i,p}(x)|$ for $i=1,2$ of the predicted solution functions $\phi_{p}$ and $c_{i,p}$ by our PNPic deep learning solver with the error functions $|\phi(x) - \phi_{h_2}(x)|$ and $|c_i(x) - c_{i,h_2}(x)|$ of the input solution functions $\phi_{h_2}$ and $c_{i,h_2}$ generated by our PNPic finite element software package on the coarse grid mesh with mesh size $h_2=1/80$. Here $\phi$ and $c_i$ denote the reference solution functions, which we generated by our PNPic finite element software package on the fine grid mesh with mesh size $h_{ref}=1/640.$