Path-independent Flow Matching for Multi-parameter Generative Dynamics

2026-05-13

AI Summary

This work addresses the limitation of existing flow matching methods, which fail to guarantee path independence under multi-parameter variations, leading to generation processes that depend on specific transport trajectories. To overcome this, the paper proposes Path-independent Flow Matching (PiFM), which learns vector fields satisfying path-independent differential constraints, thereby extending flow matching to high-dimensional parameter spaces and enabling consistent, composable transport determined solely by initial and terminal distributions. Theoretical analysis reveals an intrinsic connection between PiFM and Wasserstein barycenters, and a computationally tractable training objective is derived via multi-parameter conditional probability path regression without requiring simulation. Experiments demonstrate that PiFM outperforms current approaches on both synthetic and real-world data, generating path-independent trajectories and effectively synthesizing high-quality out-of-distribution samples.

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.

Problem

• Flow Matching
• path independence
• multi-parameter generative dynamics
• transport maps
• Wasserstein barycenter

Innovation

• Path-independent Flow Matching
• multi-parameter generative dynamics
• Wasserstein barycenter
• vector field learning
• distribution interpolation