Path-independent Flow Matching for Multi-parameter Generative Dynamics

TL;DR

This paper introduces Path-independent Flow Matching (PiFM), a novel method for learning path-independent transport maps between distributions, extending flow matching to multi-parameter domains with practical training procedures.

Contribution

PiFM generalizes flow matching to multi-parameter spaces, ensuring path independence and approximating Wasserstein barycenters with a tractable training objective.

Findings

PiFM outperforms existing methods on synthetic data.

PiFM effectively interpolates path-independent trajectories.

PiFM generates out-of-distribution samples accurately.

Abstract

Flow Matching is a powerful framework for learning transport maps between probability distributions. Yet its standard single-parameter formulation is not designed to capture multi-parameter variations where the resulting transport should be path-independent. Path independence is crucial because it ensures that transformations depend only on the initial and target distributions, not on the specific path. In this work, we introduce Path-independent Flow Matching (PiFM), a method for learning vector fields whose induced flows yield path-independent transport between distributions. We show that PiFM generalizes Flow Matching to higher-dimensional parameter domains while enforcing structural conditions that ensure consistency of composed transformations. In addition, we show that, under suitable assumptions, PiFM approximates the Wasserstein barycenter, linking the framework to a notion of distributional interpolation. To enable practical training, we propose a tractable, simulation-free objective that regresses onto multi-parameter conditional probability paths. We showcase empirically that PiFM outperforms other approaches on both synthetic and real world data in interpolating path-independent trajectories and generating desired out of distribution samples.