Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference

Abstract

Population dynamics are often observed only as unpaired snapshots, which leave the evolution between measurements underdetermined. Second-order regularization can constrain it, but a penalty on the accelerations of sampled particles is a prior on joint particle paths, and different particle representations of the same marginal curve can have different acceleration costs. We therefore introduce Neural Wasserstein Splines (NWSP), a sample-based method that learns a continuous population curve under a Wasserstein smoothness prior. A neural network maps latent variables to the control points of a cubic spline in time, so samples at any time come from the spline basis without numerical integration. We project the generator's velocity onto gradients of a finite dictionary of scalar potentials and differentiate this measure-dependent projection in time, obtaining a finite-dimensional surrogate for the intrinsic Wasserstein acceleration, trained jointly with distributional fit losses. For a fixed dictionary and ridge parameter, we prove that the population penalty depends only on the marginal curve and bound its sensitivity to moment-estimation errors. Learned potentials (NWSP-N) sharpen acceleration estimates. On various synthetic and single-cell benchmarks, NWSP is highly competitive with transport and flow-based methods, ranking in the top two in 11 of 16 settings. The penalty improves on an unregularized spline in 13 of 16 settings and, while competitive with a particle-acceleration penalty, adds provable representation invariance.