SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants
This work addresses the challenge of accurately modeling dynamical systems in generative frameworks, which is important for applications like cellular trajectory inference, though it appears incremental as it builds on existing flow matching methods with a novel interpolation technique.
The paper tackled the problem of modeling dynamical systems with flow matching, which current methods struggle with due to linear interpolants that fail to capture higher-order dynamics from irregular observations, and introduced SplineFlow using B-spline interpolation to address this, resulting in strong improvements over baselines across various systems and tasks.
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 naïve 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.