Solving Vlasov Equation with Neural Networks

The Vlasov equation in a one-dimensional formulation without electric field is solved numerically using the neural networks. The training of neural network is based on the Vlasov equation only without taking any analytical solution into account. Though Vlasov equation without electric field is very similar to convection equation, still it is important to say that exactly Vlasov equation is solved here. It is important because the electric field could be easily added to the solution method proposed in this paper. The applied method has an advantage over the finite-difference methods for solving the Vlasov equation, which is expressed in the absence of errors where the derivative of the solution has a discontinuity, and also in the fact that the neural network solution method can be transferred almost without changes to the two-dimensional and three-dimensional formulation of the problem. The neural network that solves the Vlasov equation is being trained by minimizing the loss function built on the basis of Vlasov equation.