# Quantum Dynamics via Score Matching on Bohmian Trajectories

arXiv: https://arxiv.org/abs/2604.25137 · HTML: https://mlsanity.com/paper/2604.25137/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 42 references that are in the map.
- First posted on arXiv 28 Apr 2026.

## Builds on (most cited first, 18 of 18)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)
- [Monge-Amp`ere Flow for Generative Modeling](https://mlsanity.com/paper/1809.10188/) (2018)
- [Self-Consistency of the Fokker-Planck Equation](https://mlsanity.com/paper/2206.00860/) (2022)
- [Self-Consistent Velocity Matching of Probability Flows](https://mlsanity.com/paper/2301.13737/) (2023)
- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [m^ast of two-dimensional electron gas: a neural canonical transformation study](https://mlsanity.com/paper/2201.03156/) (2022)
- [Deep Stochastic Mechanics](https://mlsanity.com/paper/2305.19685/) (2023)
- [Deep Variational Free Energy Calculation of Hydrogen Hugoniot](https://mlsanity.com/paper/2507.18540/) (2025)
- [Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2512.19196/) (2025)

## Abstract (arXiv)

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.