Quantum Dynamics via Score Matching on Bohmian Trajectories

TL;DR

This paper introduces a neural network-based method to solve the Schrödinger equation by learning the score function on Bohmian trajectories, enabling a generative modeling approach to quantum dynamics.

Contribution

It proposes a novel score matching framework on Bohmian trajectories that recovers Schrödinger dynamics for nodeless wave functions, bridging quantum mechanics and generative modeling.

Findings

Successfully modeled wavepacket splitting in a double-well potential.

Accurately captured anharmonic vibrations of a Morse chain.

Proved the method recovers Schrödinger dynamics for nodeless wave functions.

Abstract

We solve the time-dependent Schr\"odinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under the classical potential supplemented by a quantum potential depending on the score function of the evolving density. These non-crossing Bohmian trajectories form a continuous normalizing flow governed by the score. We parametrize the score with a neural network and minimize a self-consistent Fisher divergence between the network and the score of the resulting density. We prove that the zero-loss minimizer of this self-consistent objective recovers Schr\"odinger dynamics for nodeless wave functions, a condition naturally met in quantum vibrations of atoms. We demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain. By recasting real-time quantum dynamics as a self-consistent score-driven normalizing flow, this framework opens the time-dependent Schr\"odinger equation to the rapidly advancing toolkit of modern generative modeling.