# TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics

arXiv: https://arxiv.org/abs/2608.21070 · HTML: https://mlsanity.com/paper/2608.21070/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 74 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 30 of 40)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [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)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold](https://mlsanity.com/paper/2408.14608/) (2024)
- [Variational Rectified Flow Matching](https://mlsanity.com/paper/2502.09616/) (2025)
- [Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation](https://mlsanity.com/paper/2311.15100/) (2023)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [Augmented Bridge Matching](https://mlsanity.com/paper/2311.06978/) (2023)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Towards Hierarchical Rectified Flow](https://mlsanity.com/paper/2502.17436/) (2025)
- [Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modeling](https://mlsanity.com/paper/2505.13413/) (2025)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [Constant Acceleration Flow](https://mlsanity.com/paper/2411.00322/) (2024)
- [Variational Regularized Unbalanced Optimal Transport: Single Network, Least Action](https://mlsanity.com/paper/2505.11823/) (2025)
- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)

## Built on by (most cited first, 1 of 1)

- [Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference](https://mlsanity.com/paper/2610.19878/) (2026)

## Abstract (arXiv)

Inferring continuous system evolution from sparse temporal snapshots is a key challenge in generative modeling and single-cell omics. While Optimal Transport (OT) is popular, existing frameworks are largely restricted to first-order dynamics, assuming memoryless velocity fields. This limits expressiveness, as first-order systems fail to account for regulatory momentum and time-delayed responses inherent in processes like cell differentiation. Here, we introduce TracingFlow, a simulation-free Flow Matching framework generalizing to second-order dynamics. By using neural networks to regress the acceleration field, TracingFlow provides an exact, efficient solution to the Dynamical Optimal Acceleration Transport (DOAT) problem. Unlike first-order methods yielding over-smoothed trajectories, our second-order formulation captures high-curvature transitions and nonlinear evolutions by learning the underlying force fields. Evaluated on complex synthetic and large-scale scRNA-seq datasets, TracingFlow achieves superior accuracy in distributional reconstruction and trajectory faithfulness. Moreover, by integrating lineage tracing priors, it recovers dynamical structures that are both mathematically optimal and biologically plausible.