Using neural networks to solve nonlinear differential equations in atomic and molecular physics

To represent the solution of a differential equation by an artificial neural network (ANN) was an idea introduced by Lagaris. Sugawara applied this concept to solve Schrodinger's equation for select systems. We have submitted their method to a new kind of application. Here, for the first time, the approach is applied to the equations derived from density functional theory (DFT). At first, we have tested the procedure for two simple systems: the double harmonic oscillator and the hydrogen atom. The ANN solutions obtained for these simple systems reproduced the analytical results easily. Next, we have moved to the Tomas–Fermi theory and the Kohn–Sham formulation of DFT. In order to show the feasibility of the ANN representation of electronic density, we have solved the Hooke model-atom and two light atoms: helium and lithium. The ANN results match well with the analytical solution to the Hooke model-atom and with the numerical solutions for helium and lithium. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011