用神经网络学习量子轨迹的梯度，实现实时量子动力学模拟。
Quantum Dynamics via Score Matching on Bohmian Trajectories
- 通过神经网络拟合波函数概率密度的梯度（得分函数）
- 在双阱势和莫尔斯链系统中成功模拟波包分裂与非谐振动
- 将量子演化转为自洽得分驱动的生成模型，适合生成式算法研究者
我们通过在玻姆轨迹上学习得分函数（概率密度对数的梯度）来求解时变薛定谔方程。在玻姆量子力学框架中，粒子沿确定性路径运动，受经典势和依赖于得分函数的量子势共同作用。这些不交叉的玻姆轨迹构成由得分函数控制的连续归一化流。我们用神经网络参数化得分函数，并最小化网络与生成密度得分之间的自洽Fisher散度。证明了该自洽目标零损失极小值能精确恢复无节点波函数的薛定谔动力学，这一条件在原子量子振动中自然满足。我们在双阱势中的波包分裂和莫尔斯链的非谐振动上验证了该方法。通过将实时量子动力学重构为自洽得分驱动的归一化流，本框架使时变薛定谔方程可直接应用现代生成建模的快速进展工具。
原文摘要 · Abstract (English)
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.
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