Physics-informed neural networks for solving forward and inverse Vlasov–Poisson equation via fully kinetic simulation

Abstract

The Vlasov–Poisson equation is one of the most fundamental models in plasma physics. It has been widely used in areas such as confined plasmas in thermonuclear research and space plasmas in planetary magnetospheres. In this study, we explore the feasibility of the physics-informed neural networks for solving forward and inverse Vlasov–Poisson equation (PINN-Vlasov). The PINN-Vlasov method employs a multilayer perceptron (MLP) to represent the solution of the Vlasov–Poisson equation. The training dataset comprises the randomly sampled time, space, and velocity coordinates and the corresponding distribution function. We generate training data using the fully kinetic PIC simulation rather than the analytical solution to the Vlasov–Poisson equation to eliminate the correlation between data and equations. The Vlasov equation and Poisson equation are concurrently integrated into the PINN-Vlasov framework using automatic differentiation and the trapezoidal rule, respectively. By minimizing the residuals between the reconstructed distribution function and labeled data, and the physically constrained residuals of the Vlasov–Poisson equation, the PINN-Vlasov method is capable of dealing with both forward and inverse problems. For forward problems, the PINN-Vlasov method can solve the Vlasov–Poisson equation with given initial and boundary conditions. For inverse problems, the completely unknown electric field and equation coefficients can be predicted with the PINN-Vlasov method using little particle distribution data.