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

2026-01-30

AI Summary

This work addresses the challenge of accurately modeling complex state evolution in high-order dynamical systems under irregularly sampled observations—a setting where existing flow matching methods falter due to their reliance on linear interpolation. To overcome this limitation, we propose SplineFlow, the first framework to integrate B-spline basis functions into flow matching. By leveraging B-spline interpolation, SplineFlow constructs smooth and stable conditional trajectories that satisfy multi-marginal constraints, thereby capturing intricate temporal dynamics more faithfully. The method combines continuous normalizing flows with multi-marginal constrained optimization, achieving significant performance gains over current baselines across diverse tasks, including deterministic and stochastic dynamical systems as well as single-cell trajectory inference, particularly in scenarios involving high-order dynamics and irregular sampling.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: State Estimation
• Machine Learning: Time-Series/Data Streams

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• flow matching
• dynamical systems
• B-spline interpolation
• multi-marginal constraints
• irregular sampling

Innovation

• Flow Matching
• B-spline Interpolation
• Dynamical Systems
• Multi-marginal Constraints
• Continuous Normalizing Flows