A physics-informed neural network based method for the nonlinear Poisson-Boltzmann equation and its error analysis

In this work, we develop a physics-informed neural network based method to solve the nonlinear Poisson-Boltzmann (PB) equation. One challenge in predicting the solution of the PB equation arises from the Dirac-delta type singularities, which causes the solution to blow up near the singular charges. To manage this issue, we construct Green-type functions to handle the singular component of the solution. Subtracting these functions yields a regularized PB equation exhibiting discontinuity across the solute-solvent interface. To handle the discontinuities, we employ a continuous Sobolev extension for the solution of the regularized PB equation on each subdomain. By adding an augmentation variable to label the sub-regions, we are able to achieve a continuous extension of the regularized solution. Finally, the physics-informed neural network (PINN) is proposed, where the parameters are determined by a judiciously chosen loss functional. In this way, we propose a user-friendly efficient approximation for the PB equation without the necessity for any mesh generation or linearization process such as the Newton-Krylov iteration. The error estimates of the proposed PINN method are carried out. We prove that the error between the exact and neural network solutions can be bounded by the physics-informed loss functional, whose magnitude can be made arbitrarily small for appropriately trained neural networks with sufficiently many parameters. Several numerical experiments are provided to demonstrate the performance of the proposed PINN method.