Learning path splines via Acceleration Matching

Given snapshot observations of a (stochastic) process, how does one learn a smooth interpolating path that simultaneously satisfies all the marginal constraints? Natural candidates to seek are path splines, or $\msf P$-splines: probability paths which, in direct analogy with the usual cubic spline, are asked to obey marginal constraints at prescribed times. Many works investigate $\msf P$-splines from the perspective of multi-marginal Schr\"odinger bridges or flow matching, though such methods are not entirely simulation-free, unstable during training, or require extensive preprocessing. In this work, we present a novel algorithm called \emph{Acceleration Matching} (\texttt{AM}). In contrast to prior approaches, \texttt{AM} learns a conditional accelerating field using an entirely simulation-free, explicit regression objective that is free of virtually any preprocessing steps. We demonstrate on low-dimensional benchmarks that our method is competitive with, and often improves upon, leading alternatives while enjoying faster training times.