Curly Flow Matching for Learning Non-gradient Field Dynamics

TL;DR

Curly Flow Matching introduces a novel method to learn non-gradient, periodic dynamics in natural systems by solving a Schrödinger bridge problem with a non-zero drift, outperforming traditional gradient-based models.

Contribution

It presents Curly-FM, a new approach that captures non-gradient, periodic behaviors in systems, extending flow matching beyond population modeling.

Findings

Curly-FM effectively models non-gradient, periodic dynamics.

It outperforms traditional flow matching in trajectory inference.

Applicable to biological, fluid, and ocean systems.

Abstract

Modeling the transport dynamics of natural processes from population-level observations is a ubiquitous problem in the natural sciences. Such models rely on key assumptions about the underlying process in order to enable faithful learning of governing dynamics that mimic the actual system behavior. The de facto assumption in current approaches relies on the principle of least action that results in gradient field dynamics and leads to trajectories minimizing an energy functional between two probability measures. However, many real-world systems, such as cell cycles in single-cell RNA, are known to exhibit non-gradient, periodic behavior, which fundamentally cannot be captured by current state-of-the-art methods such as flow and bridge matching. In this paper, we introduce Curly Flow Matching (Curly-FM), a novel approach that is capable of learning non-gradient field dynamics by designing and solving a Schr\"odinger bridge problem with a non-zero drift reference process -- in stark contrast to typical zero-drift reference processes -- which is constructed using inferred velocities in addition to population snapshot data. We showcase Curly-FM by solving the trajectory inference problems for single cells, computational fluid dynamics, and ocean currents with approximate velocities. We demonstrate that Curly-FM can learn trajectories that better match both the reference process and population marginals. Curly-FM expands flow matching models beyond the modeling of populations and towards the modeling of known periodic behavior in physical systems. Our code repository is accessible at: https://github.com/kpetrovicc/curly-flow-matching.git