解决函数流匹配的离散化与统计一致性问题，确保有限采样下收敛性。
Discretization and Statistical Consistency of Functional Flow Matching
- 基于有限系数或点值实现函数流匹配，证明强L²收敛性。
- 给出正交投影和点传感器的定量误差界，验证常数独立于传感器分布。
- 适用于深度学习中的流模型，提供端到端的Wasserstein误差边界。
函数流匹配在函数分布上定义，但通过有限个系数或点值实现。在非均匀或自适应细化下，条件σ-代数未必嵌套，因此不能依赖鞅收敛来解释传感器极限。我们证明了对任意强一致的有限秩重构序列，有限条件速度目标在强L²意义下收敛，并为正交投影给出了量化误差界，通过正则空间扩展至点传感器。对于学习型流，直接耦合到总体超位置路径可获得端到端Wasserstein界，无需假设总体有限维ODE的唯一性。我们验证了归一化求积神经算子的传感器无关常数，包括全局Lipschitz激活下的显式幅度递推关系。一个非交换迹类高斯例子显示：投影限制下边界乘子为0，精确条件下的乘子为0.72。空间正则性-积分证书封闭了算子实现项，伯恩斯坦论证给出固定模型维度与包络下的 ilde{O}(n^{-1})过剩风险项，且一个可精确实现的截断高斯缩放特例提供了显式端到端收敛速率。
原文摘要 · Abstract (English)
Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove strong $L^2$ convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space. For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound without assuming uniqueness of the population finite-dimensional ODE. We verify sensor-independent constants for a normalized quadrature neural operator, including globally Lipschitz activations through an explicit magnitude recurrence. A noncommuting trace-class Gaussian example gives boundary multiplier $0$ under projected restriction and $0.72$ under exact conditioning. A spatial regularity--cubature certificate closes the operator-realization term, a Bernstein argument gives a $\widetilde{O}(n^{-1})$ excess-risk term for fixed model dimension and envelopes, and an exactly realizable clipped Gaussian scaling specialization yields an explicit end-to-end rate.
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