Computation of the Time-Dependent Dirac Equation with Physics-Informed Neural Networks

Abstract

We propose to compute 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 and train the network with a large number of points. Specifically, we derive PINNs-based algorithms and present some key fundamental properties of these algorithms when applied to the Dirac equations in different physical frameworks.