SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants

TL;DR

SplineFlow introduces a B-spline based flow matching method that effectively models complex dynamical systems and outperforms existing approaches in various experiments.

Contribution

It presents a novel flow matching algorithm using B-spline interpolation to better capture higher-order dynamics in dynamical systems.

Findings

SplineFlow outperforms existing methods on deterministic and stochastic systems.

The method effectively models complex underlying dynamics.

It improves cellular trajectory inference tasks.

Abstract

Flow matching is a scalable generative framework for characterizing continuous normalizing flows with wide-range applications. However, current state-of-the-art methods are not well-suited for modeling dynamical systems, as they construct conditional paths using linear interpolants that may not capture the underlying state evolution, especially when learning higher-order dynamics from irregular sampled observations. Constructing unified paths that satisfy multi-marginal constraints across observations is challenging, since na\"ive higher-order polynomials tend to be unstable and oscillatory. We introduce SplineFlow, a theoretically grounded flow matching algorithm that jointly models conditional paths across observations via B-spline interpolation. Specifically, SplineFlow exploits the smoothness and stability of B-spline bases to learn the complex underlying dynamics in a structured manner while ensuring the multi-marginal requirements are met. Comprehensive experiments across various deterministic and stochastic dynamical systems of varying complexity, as well as on cellular trajectory inference tasks, demonstrate the strong improvement of SplineFlow over existing baselines. Our code is available at: https://github.com/santanurathod/SplineFlow.