超越梯度流：从分布快照中的可辨识性与恢复

中文摘要 AI 辅助

从演化分布的快照推断动力学从根本上是不适定的：Fokker-Planck方程仅通过其得分加权散度$\nabla\cdot F+F\cdot\nabla\log\rho$约束漂移$F$，留下一个$\rho$-螺线管规范对任何单时间约束不可见。基于时间索引的输运公式无法解决这种模糊性：每个可接受的边际路径都允许一个无旋解释，最小作用重构选择该解释，而仅凭边际拟合无法区分动力学上不等价的解释。要求一个自治场解释多个边际，反而使部分隐藏环流随着$\nabla\log\rho$跨边际变化而变得可见。将瞬时Fokker-Planck源约束与快照实验分离，我们证明源约束在堆叠得分加权散度算子的核模下识别该场。对于一般高斯形状变化，在内禀维度$m$下，$K\ge m$个时间点的源约束消除每个多项式规范方向，而仅有限多个密度快照允许混叠；我们明确给出该障碍。在高斯锚点处，对于Sobolev光滑度$s$和每个时间点$n$个样本，我们推导出切线快照实验的条件较低速率$(nK)^{-2s/(2s+m+1)}$，并在逐度基准中匹配上速率。强形式拟合与得分误差非正交，且无法通过谱滤波修复。相反，我们使用光滑测试函数估计，同时保留已知扩散项，并推导出将采样误差与固定网格求积偏差分离的有限样本界。植入环流实验证实了预测的规范收缩，并暴露了跨切片信息与协方差感知白化之间的设计张力。

英文摘要

Inferring dynamics from snapshots of evolving distributions is fundamentally underdetermined: the Fokker-Planck equation constrains the drift $F$ only through its score-weighted divergence $\nabla\cdot F+F\cdot\nabla\logρ$, leaving a $ρ$-solenoidal gauge invisible to any single-time constraint. Time-indexed transport formulations cannot resolve this ambiguity: every admissible marginal path admits a curl-free explanation, minimum-action reconstruction selects it, and marginal fit alone cannot distinguish dynamically inequivalent explanations. Requiring one autonomous field to explain several marginals instead makes part of the hidden circulation visible as $\nabla\logρ$ changes across marginals. Separating instantaneous Fokker-Planck source constraints from the snapshot experiment, we show that the source constraints identify the field modulo the kernel of a stacked score-weighted divergence operator. For generic Gaussian shape variation, source constraints at $K\ge m$ time points in intrinsic dimension $m$ eliminate every polynomial gauge direction, whereas finitely many density snapshots alone admit aliasing; we give the obstruction explicitly. At a Gaussian anchor, for Sobolev smoothness $s$ and $n$ samples per time point, we derive a conditional lower rate $(nK)^{-2s/(2s+m+1)}$ for the tangent snapshot experiment, with a matching upper rate in a degreewise benchmark. Strong-form fitting is non-orthogonal to score error and cannot be repaired by spectral filtering. Instead, we estimate using smooth test functions while retaining the known diffusion term, and derive a finite-sample bound that separates sampling error from fixed-grid quadrature bias. Planted-circulation experiments confirm the predicted gauge contraction and expose a design tension between cross-slice information and covariance-aware whitening.

发表机构

• VIB, Center for Molecular Neurology（VIB分子神经学中心）
• VIB, Center for AI and Computational Biology（VIB人工智能与计算生物学中心）
• University of Antwerp（安特卫普大学）
• University of Cambridge（剑桥大学）

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