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This paper introduces a novel approach to solving the time-dependent Schr枚dinger equation by learning the score function on Bohmian trajectories, which are deterministic paths influenced by both classical and quantum potentials. By employing a neural network to parametrize the score and minimizing a self-consistent Fisher divergence, the authors demonstrate that their method recovers Schr枚dinger dynamics for nodeless wave functions. The framework is validated through applications to wavepacket splitting and anharmonic vibrations, showcasing its potential to leverage modern generative modeling techniques in quantum dynamics.
Learning quantum dynamics through Bohmian trajectories transforms the time-dependent Schr枚dinger equation into a self-consistent generative modeling problem.
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.
