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

arXiv: https://arxiv.org/abs/2610.19878 · HTML: https://mlsanity.com/paper/2610.19878/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 39 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Builds on (most cited first, 23 of 23)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Trajectory Inference with Smooth Schr"odinger Bridges](https://mlsanity.com/paper/2503.00530/) (2025)
- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)
- [Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging](https://mlsanity.com/paper/2405.19679/) (2024)
- [Wasserstein-Fisher-Rao Splines](https://mlsanity.com/paper/2203.15728/) (2022)
- [A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots](https://mlsanity.com/paper/2605.08550/) (2026)
- [Multi-Marginal Flow Matching with Adversarially Learnt Interpolants](https://mlsanity.com/paper/2510.01159/) (2025)
- [SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants](https://mlsanity.com/paper/2601.23072/) (2026)
- [Wassersplines for Neural Vector Field--Controlled Animation](https://mlsanity.com/paper/2201.11940/) (2022)
- [PACE: Geometry-Aware Bridge Transport for Single-Cell Trajectory Inference](https://mlsanity.com/paper/2605.18587/) (2026)
- [Trajectory inference via Acceleration Matching](https://mlsanity.com/paper/2608.03916/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)

## Abstract (arXiv)

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.