Improving Flow Matching by Aligning Flow Divergence

TL;DR

This paper enhances conditional flow matching by aligning flow divergence, leading to more accurate probability path learning and improved generative model performance across various benchmarks.

Contribution

Introduces a PDE-based divergence alignment method that improves flow matching accuracy in flow-based generative models.

Findings

Improved generative performance on benchmarks

Theoretical bound on divergence gap

Enhanced training objective for flow models

Abstract

Conditional flow matching (CFM) stands out as an efficient, simulation-free approach for training flow-based generative models, achieving remarkable performance for data generation. However, CFM is insufficient to ensure accuracy in learning probability paths. In this paper, we introduce a new partial differential equation characterization for the error between the learned and exact probability paths, along with its solution. We show that the total variation gap between the two probability paths is bounded above by a combination of the CFM loss and an associated divergence loss. This theoretical insight leads to the design of a new objective function that simultaneously matches the flow and its divergence. Our new approach improves the performance of the flow-based generative model by a noticeable margin without sacrificing generation efficiency. We showcase the advantages of this enhanced training approach over CFM on several important benchmark tasks, including generative modeling for dynamical systems, DNA sequences, and videos. Code is available at \href{https://github.com/Utah-Math-Data-Science/Flow_Div_Matching}{Utah-Math-Data-Science}.