Innovative Neural Network Approach for Solving TDKS Equations with Jensen’s Inequality

Abstract

We propose a novel neural network-based approach for solv-ing the Time
Dependent Kohn-Sham (TDKS) equations, central to Time-Dependent Density Functional Theory (TDDFT). Focusing on the one-electron case, where the TDKS reduces to the time-dependent Schr¨odinger equation, we employ modified Physics Informed Neural Networks (PINNs) incorporating Jensen’s inequality in place of the  traditional Mean Squared Error (MSE) loss. This convex formulation improves training  stability and solution accuracy. Compared to the classical Runge-Kutta (RK4) method, our  approach achieves comparable accuracy while demonstrating supe-rior scalability and  smoother convergence, especially for stiff or nonlinear dynamics. This work establishes a  foundation for extending neural PDE solvers to more complex quantum systems.that the  classroom teaching ex-periment began.