Time-dependent Dirac Equation with Physics-Informed Neural Networks: Computation and Properties

Abstract

In this paper, we are interested in the computation of the time-dependent Dirac equation using physics-informed neural networks (PINNs), a new powerful tool in scientific machine learning avoiding the use of approximate derivatives of differential operators. PINNs search solutions in the form of parameterized (deep) neural networks, whose derivatives (in time and space) are performed by automatic differentiation. The computational cost comes from the need to solve high-dimensional optimization problems using stochastic gradient methods in the training the network with a large number of points analogues of the discretization points for standard PDE solvers. Specifically, we derive a PINNs-based algorithm and present some key fundamental properties when applied to the Dirac equations in different physical frameworks.