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

Flow matching is an emerging, scalable generative framework for characterizing continuous normalizing flows with wide-range applications. However, state-of-the-art methods are not well-suited for modeling dynamical systems, as they construct conditional paths that are restricted to linear interpolants, a suboptimal supervision signal, which may not capture the system's underlying state evolution. Moreover, constructing unified paths to satisfy multi-marginal constraints across observations is challenging, since naïve higher-order polynomials tend to be unstable and oscillatory. To address these limitations, 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 deterministic and stochastic dynamical systems under various configurations, as well as cellular trajectory inference tasks, demonstrate that SplineFlow outperforms existing baselines, especially when the underlying dynamics are of higher degree or in irregular sampling regimes.