Trajectory inference via Acceleration Matching

Trajectory inference aims to reconstruct smooth, continuous paths from unpaired snapshots of data collected at discrete time points. Existing methods often struggle with computational efficiency, requiring either expensive simulation-based training or complex preprocessing to enforce smoothness. The authors propose Acceleration Matching (AM), which addresses these issues by lifting the problem to phase space (position and velocity). By regressing onto an explicit conditional acceleration field, AM induces smooth dynamics that naturally interpolate the prescribed marginal distributions without needing to simulate trajectories during the training process.

The core of AM is the Markovianization of kinetic Brownian bridges. The authors show that the interpolation problem can be framed as a sequence of stochastic differential equations (SDEs) where the acceleration field is learned via a neural network. Because the acceleration field is defined as a conditional expectation, the training objective is a simple regression loss that only requires positional data. At inference time, the model uses a learned conditional sampler to estimate initial velocities, followed by an Euler-Maruyama integration to generate the full trajectories.

Numerical experiments on benchmark datasets, such as ocean-current trajectories (GoM) and Lotka-Volterra (LV) dynamics, demonstrate that AM is competitive with or superior to existing state-of-the-art methods. The trajectories generated by AM are visually smoother than those produced by piecewise flow matching, and the algorithm achieves lower Wasserstein errors on held-out marginals. By avoiding simulation during training and eliminating the need for ad hoc preprocessing, AM provides a more efficient and stable framework for learning second-order dynamics.