Discretization and Statistical Consistency of Functional Flow Matching

Functional flow matching aims to learn time-dependent velocity fields on infinite-dimensional function spaces. However, practical implementations must rely on finite-dimensional observations (e.g., point values or coefficients). The author investigates whether these finite-dimensional approximations are statistically consistent—that is, whether the learned flows converge to the true continuum-space flow as the resolution of the sensors increases, particularly when the observation grids are not nested.

To address this, the paper establishes a rigorous framework for conditional expectation in Hilbert spaces. It addresses the challenge that non-nested grids prevent the use of standard martingale convergence theorems. The author proves strong L2 convergence of finite conditional velocity targets by showing that the finite-dimensional regression targets converge to the continuum target. The analysis includes quantitative bounds for orthogonal projections and extends these results to point-sensor configurations using a regularity space (Sobolev spaces). The paper also provides a stability analysis for learned flows, ensuring that the end-to-end Wasserstein distance between the generated and target distributions remains controlled.

• Target Consistency: The paper demonstrates that the finite-dimensional conditional velocity targets converge to the continuum target in L2, even without nested sigma-algebras. This provides a theoretical foundation for using adaptive or scattered sensors in functional flow matching.
• Error Bounds: The author provides explicit quantitative bounds for the convergence of these targets, including a specific analysis for orthogonal projections and point sensors.
• Stability: The study shows that the learned ODEs remain stable, allowing for an end-to-end Wasserstein bound without requiring the uniqueness of the population-level ODE.
• Counter-examples: The paper includes a cautionary result showing that a globally Lipschitz continuum velocity field does not guarantee that its finite conditional targets will have a mesh-uniform Lipschitz envelope, highlighting the necessity of verifying stability assumptions for specific neural operator architectures.