Source-linked AI summary
Learning smooth dynamics via Kinematic Interpolants
Yiheng, Jiang, Youssef Marzouk, Avishek Joey Bose, Aram-Alexandre Pooladian
TL;DR
Position-only snapshots can leave the underlying dynamics ambiguous, and existing methods have difficulty exploiting available velocity measurements scalably. Kinematic Interpolants learn acceleration fields from position–velocity snapshots, and experiments report improved reconstruction accuracy at lower training cost than compared baselines. The current formulation does not explicitly encode physical priors, and experiments largely focus on interpolation.
Problem
Position-only snapshots may be consistent with multiple dynamical systems, while incorporating increasingly available velocity measurements in scalable models remains challenging.
Source: §1 · p. 1Method
Kinematic Interpolants extend stochastic interpolants to position–velocity snapshots, learning acceleration fields with velocity defined as the position derivative.
Source: §1 · p. 2Results
Across ocean vortices, mouse erythroid differentiation, and collective bird motion, KI improves reconstruction accuracy and substantially lowers training costs relative to compared baselines.
Source: §7 · p. 13Takeaways & Limitations
KI provides a framework for reconstructing population dynamics from unpaired, time-indexed position–velocity snapshots through simulation-free acceleration-field training.
Source: §7 · p. 13The current formulation does not explicitly encode physical priors, and experiments largely focus on interpolation.
Source: §7 · p. 13
Abstract
from arXiv · showhide
Inferring dynamics from real-world experimental data is inherently ambiguous, as many plausible dynamical systems can explain the same positional observations at measurement times. For higher fidelity modeling of applications ranging from single-cell RNA to vortex dynamics and even flocking of birds, scientific observables increasingly also include velocity estimates that serve to resolve this issue. However, incorporating this rich information in a scalable manner, free of simulation or unnecessary approximations, has remained a difficult challenge. In this paper, we introduce Kinematic Interpolants (KI), a novel framework that extends the seminal methodology of stochastic interpolants to infer nonlinear dynamics across multiple snapshots in phase space, i.e., observables with position and velocity measurements. Empirically, we demonstrate that across several scientific datasets, KI achieves superior reconstruction of the underlying dynamics compared to existing baselines while requiring only a fraction of the training time.
1 Introduction
Population measurements provide unpaired snapshots, and positions alone may not identify the governing dynamics. Kinematic Interpolants use position–velocity observations to learn acceleration fields, with reported improvements in reconstruction accuracy and training cost.
1 Introduction
- Unpaired snapshots make trajectory inference a task of recovering continuous dynamics consistent with population measurements rather than tracking individuals across time.The same cell cannot be tracked between destructive single-cell RNA measurements.Source: §1 · p. 1
- Position-only observations can leave dynamics ambiguous, while velocity measurements add constraints on reconstruction.Rotationally symmetric dynamics around a fixed center can yield identical position snapshots.Source: §1 · pp. 1–2
- Existing velocity-aware methods make different trade-offs: some use approximate reference drifts or trajectory simulation, while others first impute missing velocities.The paper motivates probabilistic second-order dynamics with simulation-free training for observed position–velocity data.Source: §1 · p. 2
- Kinematic Interpolants extend stochastic interpolants to position–velocity snapshots, defining velocity as the position derivative so both remain consistent by construction.Source: §1 · p. 2
- KI improves reconstruction accuracy across several scientific datasets at substantially lower training cost than existing baselines.Its contributions include conditional-acceleration dynamics, closed-form targets for simulation-free regression, and deterministic or stochastic sampling.Source: §1 · p. 2
2 Background
The background defines dynamics on joint position–velocity phase space, where position evolves according to velocity and acceleration governs velocity change. It also describes a stochastic formulation and a velocity-score correction that matches its time marginals to those of the deterministic dynamics.
2 Background
- Phase space is the joint position–velocity space, and its density evolves according to a continuity equation on that joint space.The background assumes sufficient smoothness and finite displayed expectations and integrals.Source: §2 · pp. 2–3
- Physical second-order dynamics constrain position change to the velocity component, yielding acceleration as the derivative of velocity.The resulting density evolution is expressed by the kinetic continuity equation.Source: §2 · p. 3
- For stochastic phase-space paths, the background introduces diffusion in velocity and a corresponding kinetic Fokker–Planck equation.The acceleration field is replaced by a drift field in the stochastic dynamics.Source: §2 · p. 3
- A velocity-score correction is chosen so the stochastic dynamics have the same time marginals as the deterministic phase-space dynamics.The passage describes the correction term as canceling with the diffusion term.Source: §2 · p. 3
3 Kinematic interpolants
Kinematic interpolants construct paths through observed position–velocity distributions and use their accelerations to define dynamics. The framework specifies endpoint-consistent paths, a conditional-mean acceleration field, and choices of mean path and noise schedule.
3 Kinematic interpolants
- At irregular observation times, the task is to match joint position–velocity distributions while generating samples at intermediate times.Particle identities are unavailable across snapshots, so the learned process represents continuous trajectories for independent particles.Source: §3 · p. 3
3.1 Kinematically consistent paths
- The kinematically consistent path interpolates between coupled endpoint states and adds noise while defining velocity as the path’s time derivative.The interpolant is conditioned on endpoint pairs drawn from a coupling of the observed marginals.Source: §3.1 · p. 4
3.2 Learning dynamics via accelerations
- The conditional-mean acceleration field reproduces the interpolant’s joint position–velocity distribution at every time, including the prescribed observation marginals.The conditional expectation averages over sampled endpoints and noise given the interpolant state.Source: §3.2 · pp. 4–5
3.3 On the choice of paths
- The framework constructs a phase-space Gaussian interpolant conditioned on endpoint pairs, parameterized by its mean and covariance.Source: §3.3 · p. 5
3.3.1 Choosing the mean path
- A cubic Hermite polynomial is the unique degree-at-most-three position path satisfying the four endpoint position and velocity constraints.Its first and second derivatives provide the mean velocity and deterministic acceleration-target component in closed form.Source: §3.3.1 · p. 5
3.3.2 Choosing the noise
- The noise design controls position noise and supplies additional velocity noise; bridge noise satisfies endpoint constraints, while constant noise convolves endpoint laws with a Gaussian.The changing angle provides velocity noise not determined by positional noise alone.Source: §3.3.2 · pp. 5–6
4 Training and sampling
Training regresses acceleration networks on interpolant states and pathwise acceleration targets without simulation. The pipeline also supports stochastic sampling through score learning, endpoint coupling choices, and deterministic or stochastic samplers.
4.1 Simulation-free training algorithm
- The algorithm samples endpoint pairs and times, constructs interpolant states and pathwise accelerations, then regresses the acceleration network by weighted squared error, without simulation.Source: §4.1 · p. 6
4.2 Learning stochastic dynamics
- For stochastic sample paths, training additionally requires learning the velocity score ∇v log ρt; the acceleration-only objective instead yields continuity-equation sampling.Source: §4.2 · p. 6
4.2.1 Minibatch training
- Minibatch training can use independent endpoint coupling or minibatch optimal-transport pairing, with pairwise and minibatch OT couplings also considered in experiments.Source: §4.2.1 · p. 7
4.3 Sampling
- Deterministic sampling integrates learned acceleration from initial position-velocity data; stochastic sampling instead uses the learned score and Gaussian noise under a fixed noise schedule.Source: §4.3 · p. 7
5 Numerical Experiments
The experiments evaluate KI on vortex, Boids, and mouse erythroid benchmarks against methods including Curly-FM, WLM, and OT-CFM. They report interpolation and forecasting results under different snapshot and coupling regimes.
5.1.1 Small vortex (interpolation)
- On the small vortex, both KI variants outperform OT-CFM and OT-MFM at all four held-out times; IND-KI has the lowest mean W1 at t3 and t5.Averaged over the four times, IND-KI significantly reduces error relative to WLM, the strongest baseline under that average.Source: §5.1.1 · p. 9
5.1.2 Large vortex (forecasting)
- OT-KI achieves the lowest mean W1 for large-vortex forecasting from either start time, improving accuracy over WLM by about 14% from t0 and over 55% from t9.Forecasting predicts the positional distribution at t10, starting from t0 or t9.Source: §5.1.2 · p. 9
5.2.1 Dense snapshots with pre-computed OT
- KI-bridge has the lowest mean W1 and W2 errors across both dense Boids horizons; on the long horizon, KI is about 100× faster than WLM and 7× faster than Curly-FM.Dense experiments use precomputed OT couplings between neighboring snapshots with all 1,000 particles.Source: §5.2.1 · pp. 9–10, Figure 3 · p. 10, Table 3 · p. 11
5.2.2 Sparse snapshots with minibatch OT
- With sparse Boids snapshots and minibatch OT, KI-const has the lowest mean W1 and W2 errors among sparse methods across both horizons, although errors rise relative to dense-grid KI.Larger time gaps mitigate acceleration amplification from positional discrepancies in mismatched minibatch endpoint pairs.Source: §5.2.2 · pp. 10–11
5.3 Mouse Erythroid
- For mouse erythroid data, KI-const performs best and both KI noise schedules have lower mean W2 errors than every evaluated baseline across UMAP and 20- and 50-dimensional PCA representations.Table 2 reports hold-out error across dimensions; Figure 4 visualizes generated marginals projected onto the first two principal components.Source: Figure 4 · p. 10, §5.3 · p. 11, Table 2 · p. 11
6 Related work
Related work spans gradient-flow and second-order population models, flow and bridge matching, and higher-order generative modeling. These approaches differ in their ability to represent rotational motion, use of velocity observations, and reliance on simulation.
Learning population dynamics
- Gradient-field population models cannot represent rotational motion, including ocean-vortex flow or progression through the cell cycle.The section contrasts these models with approaches that allow non-gradient dynamics.Source: §6 · p. 11
- WLM models interacting second-order dynamics through a population-level potential learned by differentiating through simulated trajectories, while Wasserstein Lagrangian Residuals enable simulation-free training.JKOnet differentiates through JKO steps, whereas JKOnet* and Wasserstein Residuals use residual formulations.Source: §6 · pp. 11–12
Flow and bridge matching approaches
- Curly-FM constructs a reference velocity field from discrete position–velocity observations before learning an interpolant, whereas KI imposes observed endpoint velocities as interpolant boundary conditions.Curly-FM uses a prescribed kernel to construct its continuous reference velocity field.Source: §6 · p. 12
- Position-only methods include metric- or density-dependent transport, physical-time flow matching, and Schrödinger-bridge refinement fitted to simulated trajectories.Acceleration Matching samples unobserved endpoint velocities by conditioning a kinetic Brownian reference process, while TracingFlow assigns velocities using conditional trajectory means.Source: §6 · p. 12
Higher-order generative modeling
- Higher-order generative methods use phase-space stochastic bridges, augmented dynamics for diffusion sampling, or second-order flows for time-series forecasting.AGM derives acceleration targets from stochastic optimal control and associated probability-flow dynamics.Source: §6 · pp. 12–13
7 Conclusion
Kinematic Interpolants reconstruct population dynamics from unpaired, time-indexed position and velocity snapshots by directly learning acceleration fields through simulation-free training. The experiments report improved reconstruction accuracy and lower training costs than the compared baselines, while the formulation does not explicitly encode physical priors and experiments largely focus on interpolation.
7 Conclusion
- Kinematic Interpolants reconstruct population dynamics from unpaired, time-indexed snapshots of positions and velocities by directly learning acceleration fields through simulation-free training.The framework accommodates multiple noise schedules.Source: §7 · p. 13
- Experiments on ocean vortices, mouse erythroid differentiation, and collective bird motion demonstrate improved reconstruction accuracy and substantially lower training costs than the compared baselines.Source: §7 · p. 13
- The formulation does not explicitly encode physical priors, and the experiments largely focus on interpolation.Source: §7 · p. 13
A.1 Proof of Proposition 1
The appendix proves that the learned dynamics preserve the interpolant’s time marginals under the stated assumptions, and shows that a particular interpolant recovers Acceleration Matching bridge marginals.
A.1 Proof of Proposition 1
- The interpolant density satisfies the continuity equation, and uniqueness under standard smoothness assumptions gives matching dynamics with the same initial density and prescribed endpoint densities.The endpoint constraints make the interpolant density equal the known densities at the prescribed times.Source: §A.1 · p. 16
B Acceleration Matching as a special case
- Choosing the Hermite interpolant recovers Acceleration Matching’s conditional bridge marginals when conditioned on neighboring endpoints.Acceleration Matching assumes no velocity data and samples velocities during training.Source: B Acceleration Matching as a special… · p. 16
B.1 Proof of Proposition 2
The appendix establishes the conditional-law equivalence for the Acceleration Matching bridge, extends the conditioning construction to a damped kinetic process, and specifies noise behavior across experimental interval lengths.
B.1 Proof of Proposition 2
- The interpolant’s conditional Gaussian law matches the Acceleration Matching bridge through its Hermite mean and time-derivative velocity, with matching covariance.The proof identifies equality of the conditional laws in the interval interior and the common prescribed endpoints.Source: §B.1 · pp. 17–18
B.2 Damped bridge mean and covariance
- For a damped kinetic reference process, Gaussian conditioning gives a coordinatewise bridge mean in closed form and a corresponding covariance.The process uses dXt = Vt dt and dVt = −αVt dt + √ϵ dBt, with α, ϵ > 0.Source: §B.2 · p. 18
C Experimental Details
- With fixed hyperparameters, position- and velocity-noise standard deviations scale as h^3/2 and h^1/2, respectively.KI-bridge has zero endpoint noise, whereas KI-const retains nonzero endpoint noise when σ > 0.Source: C Experimental Details · p. 19
C.1 Oceans (Gulf of Mexico)
This appendix specifies data splits, model and sampling settings, and evaluation procedures for the vortex, Boids, and mouse erythroid experiments. It also describes the velocity preprocessing used for the erythroid experiments and the computational factors affecting Boids training time.
C.1 Oceans (Gulf of Mexico)
- The Gulf of Mexico datasets contain nine snapshots of 111 particles and eleven of 400, with separate interpolation and forecasting holdouts.Positions and velocities are two-dimensional; time and velocities are rescaled consistently.Source: §C.1 · p. 19
- Vortex KI uses cubic Hermite mean paths, rotating Gaussian noise, separate acceleration and velocity-score networks, and score-corrected SDE sampling.OT-KI recomputes minibatch OT couplings, whereas IND-KI pairs minibatches independently; OT-CFM is a position-only comparator with the same network size and training budget.Source: §C.1 · p. 19, Table 8 · p. 22
- Small-vortex rollouts start from observed positions and velocities, while large-vortex forecasts start at t0 or t9 with independent position and velocity perturbations.The evaluation compares predicted and held-out position clouds using 1-Wasserstein distance and reports means and sample standard deviations over three seeds.Source: §C.1 · p. 19
C.2 Boids
- Boids training fixes step budgets across horizons, uses 30,000 WLM steps, and gives Curly-FM and KI batch size 256 while switching to minibatch OT in sparse settings.Curly-FM uses 5,000 geometry-training steps followed by 5,000 flow-training steps; KI samples one time from every interval.Source: §C.2 · p. 20
- KI builds Boids cost matrices with a closed-form acceleration cost, while Curly-FM evaluates a learned geometry network and nearest-neighbor velocity prior for candidate endpoint pairs.Rebuilding Curly-FM's matrix at each flow-training step increases computational cost, and geometry-network size affects runtime.Source: §C.2 · p. 20, Table 5 · p. 20
C.3 Mouse Erythroid
- Mouse erythroid experiments use graph-based scVelo velocity projections and globally rescale embedded RNA velocities using mean displacement between training marginals per unit time.The baselines are rerun with the same graph-based projections for consistent preprocessing.Source: §C.3 · p. 21
- The erythroid filtered velocity mixes a Gaussian-weighted average of nearest-neighbor velocities with noise, increasing the noise weight farther from observations.The distance threshold is 5 and sigmoid steepness is 3; 2D UMAP experiments omit noise mixing.Source: §C.3 · p. 21
- Curly-FM cosine distance measures velocity alignment, with k and the evaluation time interval set separately for 2D UMAP and 20D or 50D PCA experiments.Lower cosine distance indicates better alignment.Source: §C.3 · p. 21