Quantum Dynamics via Score Matching on Bohmian Trajectories
This work provides a new method for simulating quantum dynamics by leveraging generative modeling techniques, which could benefit computational quantum mechanics.
The authors solve the time-dependent Schrödinger equation by learning the score function on Bohmian trajectories, using a neural network to minimize a self-consistent Fisher divergence. They demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain, showing that the zero-loss minimizer recovers Schrödinger dynamics for nodeless wave functions.
We solve the time-dependent Schrödinger 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 Schrödinger 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 Schrödinger equation to the rapidly advancing toolkit of modern generative modeling.