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This paper establishes strong $L^2$ convergence for finite conditional velocity targets in functional flow matching, addressing challenges posed by non-nested conditioning sigma-algebras under adaptive refinement. The authors derive quantitative bounds for orthogonal projections and extend results to a point-sensor framework, demonstrating an end-to-end Wasserstein bound for learned flows without requiring uniqueness in the population's finite-dimensional ODE. Notably, they provide sensor-independent constants for a normalized quadrature neural operator and present a Bernstein argument yielding a $\widetilde{O}(n^{-1})$ excess-risk term, enhancing the theoretical foundation of functional flow matching.
Strong $L^2$ convergence in functional flow matching reveals that learned flows can achieve robust performance even without uniqueness assumptions in their underlying dynamics.
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.
