QH-GEM：量子流体动力学生成建模
QH-GEM: Quantum-Hydrodynamic Generative Modeling
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中文总结 AI 辅助
本文提出QH-GEM框架，基于薛定谔方程Madelung形式化构建确定性受物理约束的生成模型，推导相关动力学与条件，通过数值实验验证其有效性。
中文摘要 AI 辅助
本文中，我们开发了一种基于自由粒子薛定谔方程的Madelung形式化的确定性、受物理约束的生成框架。参考玻恩概率密度与可控初始相位函数作为自由Madelung系统的初始数据，该系统通过玻姆量子势耦合玻恩概率密度与相位函数，而相位函数决定流体动力学速度场。若玻恩概率密度保持正值且流体动力学速度场生成唯一全局特征流，则从参考密度抽取并沿特征流传输的样本在每一时刻均按演化的玻恩概率密度分布。因此，随机性仅通过初始抽样引入，后续生成过程是确定性的，既不涉及随机动力学，也不包含独立参数化的时变速度场。我们将终端时刻分布匹配表述为PDE约束的相位识别问题，并推导其 underlying哈密顿量与费舍尔信息结构。对于各向同性高斯波包，我们获得了显式动力学，以及二次初始相位函数精确可达各向同性高斯目标的充要条件，同时给出了对应的抽样映射。对于光滑给定的势初始速度场，我们进一步证明特征流以O(T²)误差近似相关的一阶传输映射，该误差在均匀范数、1-瓦瑟斯坦距离与2-瓦瑟斯坦距离下均成立。数值高斯基准验证了全离散正向求解器，而针对非对称双峰目标的全网格PDE约束相位识别也得到了演示。
英文摘要
In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schrödinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time. As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.
