On the solution of Euclidean path integrals with neural networks

Abstract: This paper proposes a numerical method using neural networks to solve the path integral problem in quantum mechanics for arbitrary potentials. The method is based on a radial basis function expansion of the interaction term that appears in the Euclidean path integral formalism. By constructing a corresponding multi-layered perceptron-type neural network with exponential nonlinearities in the hidden layer, the original path integral can be approximated by a linear combination of Gaussian path integrals that can be solved analytically. The method has been tested for the double-well potential that includes a quadratic and a quartic term, giving very good, within a few percent agreement between the true and estimated bound state wave functions that are extracted from the propagator at large Euclidean times. The proposed method can also be used to describe potentials that have imaginary parts, which is tested for a simple Gaussian path integral with complex frequencies, where the model uncertainty stays below one percent for both the real and imaginary parts of the propagator.