Machine learning quantum mechanical ground states based on stochastic mechanics

he Rayleigh-Ritz variation principle is a proven way to find ground states and energies for bound quantum systems in the Schrodinger picture. Advances in machine learning and neural networks make it possible to extend it from an analytical search from a subspace of the complete Hilbert space to the a numerical search in the almost complete Hilbert space. In this paper, we extend the Rayleigh-Ritz principle to Nelson's stochastic mechanics formulation of nonrelativistic quantum mechanics and propose an algorithm to find the osmotic velocities u(x), which contain the information of a quantum systems in this picture. As a proof of concept, we calculated u(x) for one-dimensional systems, the harmonic oscillator, the double well and the Poschl-Teller potential. To obtain exited states, we calculate ground states of supersymmetrical partner Hamiltonians for each of these potentials. We will show that this method is more efficient than the stochastic optimal control algorithm that was the usual method to obtain osmotic velocities without going back to the Schrodinger equation.