Approximating a Low-Dimensional Nonlinear Dual to the Schrödinger Equation Using Neural Networks

Unlike classical physics, where the physical systems evolve in a finite-dimensional space and are typically nonlinear, quantum mechanics takes place in an infinite-dimensional space where time evolution is linear. Koopman operator theory uses an infinite dimensional operator to evolve a finite-dimensional, nonlinear dynamical system linearly in an infinite-dimensional space. We conjecture that the Schrödinger equation can be represented as the Koopman operator of a finite-dimensional system with nonlinear dynamics. Since the Koopman operator has no analytic inverse in general, we use neural networks to approximate the functions which convert the system from finite-dimensional to infinite-dimensional as well as the dynamics within the finite-dimensional space. We treat the infinite-dimensional quantum state space as observations on the finite-dimensional space and learn a basis of observations for the quantum state space. We then learn the dynamics of the compressed states, which when observed in the quantum state space reproduce the dynamics of the Schrödinger equation. Based on preliminary results on a generic quantum system, we expect that this method will be effective at replicating the dynamics of quantum systems with continuous variables.