A Deep Neural Networks-Based Method For Solving Poisson-Boltzmann Equation

The Poisson-Boltzmann equation(PBE)is a model which is widely used in the calculation of electrostatic potential for a biomolecule in an ionic solvent.In this thesis,we propose a mesh-free numerical method for solving PBE based on deep neural networks.To deal with the singularity of the PBE solution due to the Dirac-delta distribution,we decompose the solution of the PBE model only in the solute region where singularities exist,and then develop the method for solving an elliptic interface problem;to capture the strong nonlinearity in the model,we adopt a deep neural network to fit its solution,and in order to avoid the network degradation issue and the gradient vanishing or exploding issue caused by the depth of networks,residual networks together with the Xavier initialization and Swish activation function are used in the construction of the neural networks.Meanwhile,to solve the elliptic problem with discontinuous coefficient,we obtain the equivalent variational problem by the Nitsche method,and use two neural networks to fit the solutions of different coefficient regions to optimize the variational problem.Finally,to verify the feasibility,effectiveness and transferability of the algorithm,we conduct numerical experiments on a 3D PBE test model,where the relative error reaches 1.3e-2.Then we apply the method to 4 protein cases and calculate their electrostatic solvation free energies,and comparing to ones obtained using the finite element method,the relative errors are all around 1e-2.