Trajectory inference via Acceleration Matching

Trajectory inference is a fundamental problem in many scientific domains: given a collection of unpaired snapshots of observations at discrete time points, the goal is to generate smooth trajectories that best resemble and interpolate the data. Existing numerical approaches to this problem have various computational burdens, such as explicitly computing spline constructions or simulating trajectories during training. In order to overcome these limitations, we propose a new algorithm called Acceleration Matching (AM). Our approach consists of lifting the original interpolation problem to phase space and then regressing onto an explicit conditional acceleration field that induces random trajectories with smooth positional paths that, under suitable assumptions, agree with the prescribed marginals. Importantly, our resulting training algorithm only requires positional data, avoids trajectory simulation during training, and uses direct Gaussian conditioning to sample training states and targets. We provide ample numerical evidence suggesting that AM is competitive with or superior to existing algorithms on several benchmark problems from the existing literature.