Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching

Learning to transform conditional probability densities over time is a fundamental challenge spanning probabilistic modeling and the natural sciences. This task is paramount when forecasting the evolution of stochastic nonlinear dynamical systems in biological and physical domains. While flow-based models can predict the temporal evolution of probability distributions, existing approaches often assume discrete conditioning with samples that are paired across time, limiting their scientific applicability where frequently only sparse data with unpaired continuous conditioning is available. We propose Conditional Variable Flow Matching (CVFM), a framework for learning flows transforming conditional distributions with amortization across the continuous space of conditional densities. CVFM addresses the high-variance instability of prior methods by jointly sampling flows over state and conditioning variables, utilizing a conditioning mismatch kernel alongside a conditional Wasserstein distance to reweight the conditional optimal transport objective. Collectively, these advances allow for learning dynamics from sparse unpaired measurements of state-condition across time. We evaluate CVFM on conditional mapping benchmarks and a case study modeling the temporal evolution of materials internal structure during manufacturing processes, observing improved performance and convergence characteristics over existing conditional variants. Code is available at https://github.com/agenerale/conditional-variable-flow-matching.