Functional Flow Matching Consistency Proved for Discretized Data

The paper addresses a fundamental theoretical challenge in Functional Flow Matching (FFM), a technique that operates on distributions of functions but is practically implemented using a finite number of coefficients or point values. The core issue is proving that this discretization maintains statistical consistency, especially when sensor limits or refinement strategies are not nested, which can break traditional martingale convergence arguments. The researchers demonstrate strong L2 convergence of finite conditional velocity targets for any strongly consistent sequence of finite-rank reconstructions. Quantitative bounds are provided for orthogonal projections, and the framework is extended to point-sensor scenarios through a regularity space. For learned flows, the study establishes an end-to-end Wasserstein bound by directly coupling to a population superposition path, circumventing the need to assume uniqueness of the population finite-dimensional ODE. The work verifies sensor-independent constants for a normalized quadrature neural operator and provides an explicit end-to-end rate for a specific clipped Gaussian scaling specialization, offering crucial theoretical grounding for the practical application of FFM.