TracingFlow Infers System Trajectories Using Second-Order Dynamics

Inferring the continuous evolution of complex systems from limited, discrete temporal data points is a significant challenge in fields like generative modeling and single-cell omics. Current approaches, often based on Optimal Transport (OT), typically rely on first-order dynamics, assuming that system velocities are memoryless. This simplification limits their ability to accurately model processes with inherent momentum or time-delayed responses, such as cell differentiation. Researchers have introduced TracingFlow, a novel simulation-free Flow Matching framework that generalizes to second-order dynamics. Instead of just modeling velocity, TracingFlow uses neural networks to regress the acceleration field, providing an exact and efficient solution to the Dynamical Optimal Acceleration Transport (DOAT) problem. This second-order formulation allows the framework to capture more complex, high-curvature transitions and nonlinear evolutions by learning the underlying force fields that drive the system. Evaluations on both synthetic and large-scale single-cell RNA sequencing (scRNA-seq) datasets demonstrated TracingFlow's superior accuracy in reconstructing data distributions and faithfully representing trajectories. Furthermore, by incorporating lineage tracing priors, it can recover dynamic structures that are both mathematically optimal and biologically plausible, offering a more expressive and accurate tool for understanding system evolution.