Curly Flow Matching for Learning Non-gradient Field Dynamics

2025-10-30

AI Summary

Existing modeling approaches grounded in the principle of least action are restricted to gradient-field dynamics, rendering them inadequate for characterizing real-world non-gradient, periodic systems—such as single-cell RNA-based cell-cycle progression, ocean currents, and turbulent flows. Method: We propose Curly-FM, the first flow-matching framework extended to non-gradient vector fields. It constructs a reference stochastic process with non-zero drift and solves a generalized Schrödinger bridge problem to explicitly model curl-driven dynamical evolution. Curly-FM integrates population-level snapshot data with velocity inference, without assuming potential functions or time-reversibility. Contribution/Results: On single-cell trajectory inference, computational fluid dynamics, and ocean current modeling, Curly-FM consistently outperforms state-of-the-art methods—accurately recovering dynamic paths, periodic structures, and marginal distributions. It establishes a new paradigm for modeling non-equilibrium, non-gradient complex systems.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 Schrödinger 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

Problem

• Modeling transport dynamics from population-level observations in natural sciences
• Learning non-gradient field dynamics for systems with periodic behavior
• Solving trajectory inference problems for systems like cell cycles and fluid dynamics

Innovation

• Curly-FM learns non-gradient field dynamics
• Solves Schrödinger bridge with non-zero drift
• Uses inferred velocities and snapshot data