Implementation of Kolmogorov Flow Matching in a GFlowNet Learning Model

Sampling from high-dimensional probability distributions remains a major challenge due to the difficulty of estimating partition functions. Recently proposed Generative Flow Networks (GFlowNets) provide a new method to sample compositional structures efficiently. Standard GFlowNet algorithms solve the sampling problem by assuming an equilibrium model and finding a flow function that satisfies detailed balance. As such, these traditional training objectives depend on estimating forward and backward transition probabilities, flow functions, and the partition function.

We present a new Kolmogorov flow objective function, which bypasses this limitation by enforcing cycle consistency between forward and backward transition probabilities.  This approach allows us to learn equilibrium flow rates without the need for normalization from the partition function, while still yielding stable learning dynamics and performance comparable to trajectory balance, the current preferred learning objective.  This new objective thus establishes a bridge between statistical mechanics and generative modeling, and has promising applications to energy-based and entropy-regularized systems.