Learning smooth dynamics via Kinematic Interpolants

Yiheng; Jiang; Youssef Marzouk; Avishek Joey Bose; Aram-Alexandre Pooladian

cs.LGOct 15, 2026

Learning smooth dynamics via Kinematic Interpolants

Authors: Yiheng, Jiang, Youssef Marzouk, Avishek Joey Bose, Aram-Alexandre Pooladian

Organizations: Roger · Massachusetts Institute of Technology. · Imperial College London and Mila – Québec AI Institute. · Institute for Foundations of Data Science, Yale University.

Abstract

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.

Figures & tables

	Interpolation (small GoM)				Forecast (large GoM)
Method	$t_{1}$	$t_{3}$	$t_{5}$	$t_{7}$	$t_{10}$ (from $t_{0}$ )	$t_{10}$ (from $t_{9}$ )
Vanilla-SB * ( Shen et al., 2025 )	$0.112\!\pm\!0.061$	$0.178\!\pm\!0.073$	$0.262\!\pm\!0.043$	$0.305\!\pm\!0.032$	—	—
OT-CFM* ( Tong et al., 2024 )	$0.148\!\pm\!0.004$	$0.227\!\pm\!0.008$	$0.191\!\pm\!0.012$	$0.250\!\pm\!0.018$	—	—
OT-MFM* ( Kapuśniak et al., 2024 )	$0.107\!\pm\!0.014$	$0.056\!\pm\!0.014$	$0.052\!\pm\!0.011$	$0.070\!\pm\!0.021$	—	—
WLM* ( Guan et al., 2026 )	$\bm{0.039\!\pm\!0.002}$	$0.028\!\pm\!0.000$	$0.083\!\pm\!0.002$	$\bm{0.050\!\pm\!0.004}$	$0.180\!\pm\!0.038$	$0.065\!\pm\!0.005$
OT-KI (Ours)	$0.0691\!\pm\!0.0039$	$0.0224\!\pm\!0.0006$	$0.0278\!\pm\!0.0017$	$0.0639\!\pm\!0.0019$	$\bm{0.1552\!\pm\!0.0167}$	$\bm{0.0281\!\pm\!0.0066}$

Table 1: Averaged

W_{1}

distance for the GoM dataset; lower is better (*Prior results taken from Guan et al. (2026) )

Figure 1: Trajectory rollouts for GoM (small vortex).

Figure 2: Left: Boids trained on dense snapshots. Right: Boids trained on sparse snapshots.

Figure 3: Training time for long-horizon Boids experiments on a single NVIDIA L40S GPU. Both Curly-FM and KI use precomputed full-population OT in the dense regime (51 training marginals) and per-step minibatch OT (mOT) in the sparse regime (11 training marginals). More details on training budgets are listed in Table 5 .

Figure 4: First two principal components of the generated marginal for 20-d mouse erythroid, comparing constant noise schedule KI using independent couplings with the ground truth and other methods.

	$d=2$		$d=20$		$d=50$
Method	CosSim $\downarrow$	$W_{2}\downarrow$	CosSim $\downarrow$	$W_{2}\downarrow$	CosSim $\downarrow$	$W_{2}\downarrow$
OT-CFM	$0.119\pm 0.001$	$0.611\pm 0.006$	$0.326\pm 0.007$	$6.268\pm 0.052$	$0.506\pm 0.005$	$7.888\pm 0.035$
Curly-FM	$\mathbf{0.009\pm 0.000}$	$0.369\pm 0.090$	$0.335\pm 0.001$	$5.891\pm 0.003$	$0.574\pm 0.001$	$7.733\pm 0.011$
WLM	$0.038\pm 0.003$	$0.416\pm 0.026$	$0.373\pm 0.067$	$7.433\pm 0.383$	$0.522\pm 0.041$	$9.011\pm 0.173$
KI-bridge	$0.062\pm 0.003$	$0.288\pm 0.038$	$0.267\pm 0.014$	$5.490\pm 0.203$	$0.409\pm 0.016$	$7.130\pm 0.060$
KI-const	$0.072\pm 0.003$	$\mathbf{0.253\pm 0.006}$	$\mathbf{0.234\pm 0.003}$	$\mathbf{5.271\pm 0.128}$	$\mathbf{0.373\pm 0.009}$	$\mathbf{6.804\pm 0.007}$

Table 2: Hold-out error for the mouse erythroid dataset in varying dimension.

	$[0,24.5]$ (dense)		$[0,50]$ (dense)
Method	$W_{1}\downarrow$	$W_{2}\downarrow$	$W_{1}\downarrow$	$W_{2}\downarrow$
WLM	$0.594\pm 0.188$	$0.816\pm 0.209$	$1.932\pm 0.529$	$2.268\pm 0.542$
Curly-FM	$0.995\pm 0.173$	$2.424\pm 0.255$	$2.180\pm 0.254$	$3.195\pm 0.818$
KI-bridge	$\mathbf{0.292\pm 0.033}$	$\mathbf{0.476\pm 0.032}$	$\mathbf{0.450\pm 0.062}$	$\mathbf{0.682\pm 0.076}$

Table 3: Boids interpolation error with dense observations and full-data OT for KI.

	$[0,24.5]$ (sparse)		$[0,50]$ (sparse)
Method	$W_{1}\downarrow$	$W_{2}\downarrow$	$W_{1}\downarrow$	$W_{2}\downarrow$
WLM	$0.639\pm 0.036$	$0.870\pm 0.019$	$1.813\pm 0.494$	$2.131\pm 0.552$
Curly-FM	$1.476\pm 0.108$	$2.490\pm 0.199$	$2.564\pm 0.294$	$2.923\pm 0.366$
KI-const	$\mathbf{0.444\pm 0.066}$	$\mathbf{0.684\pm 0.098}$	$\mathbf{0.524\pm 0.025}$	$\mathbf{0.789\pm 0.050}$

Table 4: Boids interpolation error with sparse observations and using minibatch OT for KI.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

	Dense				Sparse
	Short		Long		Short		Long
Method	Steps	Time (min)	Steps	Time (min)	Steps	Time (min)	Steps	Time (min)
WLM	$30,000$	$276.3$	$30,000$	$361.2$	$20,000$	$107.8$	$20,000$	$203.9$
Curly-FM	$10,000$	$22.44$	$10,000$	$22.98$	$10,000$	$35.72$	$10,000$	$72.87$
KI	$20,000$	$3.10$	$20,000$	$3.27$	$20,000$	$7.86$	$20,000$	$15.13$

Appendix

Table 5: Boids training budgets and wall-clock times on a single NVIDIA L40S GPU and 8 CPUs, excluding unrelated setup and evaluation. Curly-FM step counts combine geometry and flow training. KI and Curly-FM timings include OT precomputation in the dense regime and per-step minibatch OT in the sparse regime.

Range	Coupling	Training times (s)	Number
Short	Full-batch OT	$\{0,0.5,\ldots,24.5\}$	50
Short	Minibatch OT	$\{0,5,10,15,20,24.5\}$	6
Long	Full-batch OT	$\{0,1,\ldots,50\}$	51
Long	Minibatch OT	$\{0,5,\ldots,50\}$	11
Short-range test times		$\{0.25,0.75,\ldots,24.25\}$	49
Long-range test times		$\{0.5,1.5,\ldots,49.5\}$	50

Appendix

Table 6: Training and held-out evaluation marginals for the Boids experiments. Within each time range, both coupling strategies are evaluated on the same test marginals. Each marginal contains

N=1000

particles.

	KI		Curly-FM
Hyperparameter	Bridge	Constant	Dense	Sparse
OT coupling	Full-batch OT	Minibatch OT	Full-batch OT	Minibatch OT
OT coupling population	$1000$	$256$	$1000$	$256$
Coupling construction	Precomputed	Per update	Precomputed	Per update
Sampling method	Score-corrected SDE	Score-corrected SDE	ODE	ODE
MLP hidden width	$512$	$512$	$256$	$256$

Appendix

Table 7: Boids model and training settings. Curly-FM width, depth, and learning rate apply to both geometry and flow training. KI bridge settings correspond to the 50-second long-poly run.

Hyperparameter	OT-KI	IND-KI
Endpoint coupling	Minibatch OT	Independent
Coupling construction	Recomputed per update	Resampled per update
Endpoint batch size	$128$	$128$
Mean path	Cubic Hermite	Cubic Hermite
Noise schedule	Constant amplitude	Constant amplitude
Frequency parameter $\omega$	$\pi\sqrt{3}$	$\pi\sqrt{3}$

Appendix

Table 8: Model and training settings shared across both vortex datasets for GoM.

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Trajectory inference via Acceleration Matching

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Trajectory inference is a fundamental problem in many scientific domains: given a collection of unpaired snapshots of observations at discrete time points, the goal is to generate smooth trajectories that best resemble and interpolate the data. Existing algorithms exhibit computational challenges: they either rely on preprocessing subroutines to enforce smoothness or on simulation-based training objectives, both of which can be expensive. In order to overcome these limitations, we propose a new algorithm called Acceleration Matching (AM). Our approach consists of lifting the original interpolation problem to phase space and then regressing onto an explicit conditional acceleration field that induces random, smooth trajectories that agree with the prescribed marginals. Importantly, our resulting training algorithm only requires positional data, avoids trajectory simulation during training, and is devoid of expensive preprocessing. We provide ample numerical evidence suggesting that AM is competitive with or superior to existing algorithms on several benchmark problems from the existing literature.

Abstract

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