Machine learning Kohn-Sham potentials in time-dependent density functional theory

The exact time-dependent Kohn-Sham potentials are not available due to the difficulty of approximating the exchange-correlation functional of TDDFT. In an effort to understand this approximation, we have developed a machine learning based method to obtain the Kohn-Sham potentials given the time-dependent density.  We approach this potential inversion problem by rewriting the Kohn-Sham equations as classical Hamilton’s equations. 

From the classical Hamilton’s equations, a neural network is trained using the exact time-evolved density.  The constructed neural network gives the Kohn-Sham energy functional and with it the exchange-correlation functional.  We take the advantage of the differentiable nature of the neural network to compute the necessary Kohn-Sham potential under the adiabatic approximation. We have performed numerical tests on a one-dimensional two-electron system to investigate numerical instabilities in our potential inversion method and explore the consequences of the adiabatic approximation.