SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants

Santanu Subhash Rathod; Pietro Liò; Xiao Zhang

cs.LGJan 30, 2026

SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants

Authors: Santanu Subhash Rathod, Pietro Liò, Xiao Zhang

Organizations: CISPA Helmholtz Center for Information Security, Saarbrücken, Germany · Department of Computer Science and Technology, University of Cambridge, UK

Abstract

Flow matching is a scalable generative framework for characterizing continuous normalizing flows with wide-range applications. However, current state-of-the-art methods are not well-suited for modeling dynamical systems, as they construct conditional paths using linear interpolants that may not capture the underlying state evolution, especially when learning higher-order dynamics from irregular sampled observations. Constructing unified paths that satisfy multi-marginal constraints across observations is challenging, since naïve higher-order polynomials tend to be unstable and oscillatory. We introduce SplineFlow, a theoretically grounded flow matching algorithm that jointly models conditional paths across observations via B-spline interpolation. Specifically, SplineFlow exploits the smoothness and stability of B-spline bases to learn the complex underlying dynamics in a structured manner while ensuring the multi-marginal requirements are met. Comprehensive experiments across various deterministic and stochastic dynamical systems of varying complexity, as well as on cellular trajectory inference tasks, demonstrate the strong improvement of SplineFlow over existing baselines. Our code is available at: https://github.com/santanurathod/SplineFlow.

Figures & tables

Panel of Figure 1 — Figure 1: Visualizations of interpolated trajectories from a damped harmonic oscillator.

$\bm{p}$	Model	Exp.	Harm.	Damp.	LV	Hopper.	Lorenz
$0$	NeuralODE	$9.99\mathrm{e}{-1}$	$6.54$	$3.47$	$1.98\mathrm{e}{4}$	$2.95$	$1.49$
	LatentODE	$1.00\mathrm{e}{-4}$	$8.20\mathrm{e}{-4}$	$1.20\mathrm{e}{-4}$	$\mathbf{1.56\mathrm{e}{-3}}$	$1.89$	$1.16$
	MMFM	$1.40\mathrm{e}{-3}$	$1.40\mathrm{e}{-2}$	$2.00\mathrm{e}{-3}$	$4.87\mathrm{e}{-1}$	$1.82\mathrm{e}{+1}$	$8.02\mathrm{e}{-1}$
	TFM	$6.60\mathrm{e}{-4}$	$1.81\mathrm{e}{-1}$	$1.50\mathrm{e}{-2}$	$1.84\mathrm{e}{-1}$	$1.55$	$2.44$
	SplineFlow	$\mathbf{8.90\mathrm{e}{-5}}$	$\mathbf{3.70\mathrm{e}{-4}}$	$\mathbf{9.50\mathrm{e}{-5}}$	$8.60\mathrm{e}{-2}$	$\mathbf{1.41}$	$\mathbf{6.39\mathrm{e}{-1}}$
$0.25$	LatentODE	$1.00\mathrm{e}{-4}$	$1.00\mathrm{e}{-3}$	$3.40\mathrm{e}{-4}$	$\mathbf{9.40\mathrm{e}{-4}}$	$1.61$	–

Table 1: Comparisons of different methods for modeling ODE dynamics in MSE across

6

deterministic systems and varying sampling irregularity

p

. The best performance is highlighted in bold.

Dataset	$\bm{p}$	Model	MSE (SDE)	$\bm{\mathcal{W}_{2}}$	MMD
Exp-Decay	$0$	SF2M	$0.477\pm 0.010$	$\mathbf{0.305\pm 0.005}$	$\mathbf{0.024\pm 0.001}$
	$0$	SplineFlow ( $m=1$ )	$\mathbf{0.456\pm 0.008}$	$0.331\pm 0.003$	$0.027\pm 0.002$
	$0.5$	SF2M	$\mathbf{0.524\pm 0.011}$	$0.204\pm 0.012$	$0.009\pm 0.002$
	$0.5$	SplineFlow ( $m=1$ )	$0.530\pm 0.007$	$\mathbf{0.195\pm 0.001}$	$\mathbf{0.007\pm 3.3{e}{-4}}$
Damped Harmonic	$0$	SF2M	$\mathbf{0.967\pm 0.033}$	$0.430\pm 0.007$	$0.048\pm 0.003$
	$0$	SplineFlow ( $m=1$ )	$0.970\pm 0.009$	$\mathbf{0.410\pm 0.004}$	$\mathbf{0.042\pm 0.002}$

Table 2: Comparisons of SplineFlow and SF2M for modeling SDE dynamics across multiple datasets and metrics, where

p

denotes the degree of irregularity. The best results are highlighted in bold.

Panel of Figure 2 — Figure 2: (a) and (b) compare the Lorenz ODE trajectories learned by SplineFlow with TFM, whereas (c) presents the ablations of spline degree $m$ on ODE dynamics with varying sampling irregularity $p$ .

Dataset	Task	Model	MMD	$\bm{\mathcal{W}_{2}}$	Energy
Brain Regeneration	Inter.	MOTFM	$0.503\pm 0.016$	$0.794\pm 0.026$	$1.440\pm 0.062$
		SF2M	$0.457\pm 0.008$	$0.753\pm 0.026$	$1.310\pm 0.036$
		SplineFlow $(m=5)$	$\mathbf{0.285\pm 0.007}$	$\mathbf{0.607\pm 0.027}$	$\mathbf{0.815\pm 0.021}$
	Extra.	MOTFM	$1.010\pm 0.012$	$1.570\pm 0.005$	$3.890\pm 0.051$
		SF2M	$0.972\pm 0.009$	$\mathbf{1.530\pm 0.047}$	$3.770\pm 0.072$
		SplineFlow $(m=4)$	$\mathbf{0.955\pm 0.009}$	$1.536\pm 0.019$	$\mathbf{3.740\pm 0.042}$

Table 3: Comparison results of SplineFlow on transcriptomic datasets with PHATE embedding.

Appendix figures & tables62 assets

Supplementary material from the paper’s appendix.

Appendix

Interpolant	Mathematical Form	Locality	Smoothness	Derivatives
Linear	Piecewise linear	Local	$C^{0}$	Discontinuous
Cubic	Piecewise cubic	Global	$C^{2}$	Continuous
Lagrangian	Global polynomial	Global	Smooth	Unstable
Chebyshev	Polynomial in $T_{k}$ basis	Global	Smooth	Stable
Hermite	Value+Derivative interpolation	Local	$C^{1+}$	Noise-Sensitive
B-spline	$\sum_{k}c_{k}B_{k,m}$	Local	Degree-Controlled	Stable

Appendix

Table 4: Comparison of different interpolation operators and their structural properties.

System	Linear	Non-linear	Oscillatory	Chaotic
Exponential Decay	✓	✗	✗	✗
Harmonic Oscillator	✓	✗	✓	✗
Damped Harmonic Oscillator	✓	✗	✓	✗
Lotka–Volterra	✗	✓	✓	✗
Lorenz System	✗	✓	✗	✓

Appendix

Table 5: Summary of the properties of different dynamical systems considered in our experiments.

Dynamical System	ODE	SDE (additive noise)	Model Use Case
Exponential Decay	$\dot{x}(t)=-\lambda x(t)$	$dX_{t}=-\lambda X_{t}\,dt+\sigma\,dW_{t}$	First-order relaxation processes
			radioactive decay, simple stable linear systems
			mean-reverting stochastic dynamics (OU)
Harmonic	$\dot{x}(t)=v(t)$	$dX_{t}=V_{t}\,dt$	Oscillations near stable equilibria
Harmonic	$\dot{v}(t)=-\omega^{2}x(t)$	$dV_{t}=-\omega^{2}X_{t}\,dt+\sigma\,dW_{t}$	LC circuits, linearized mechanical vibrations
Damped Harmonic	$\dot{x}(t)=v(t)$	$dX_{t}=V_{t}\,dt$	Damped mechanical vibrations

Appendix

Table 6: Full mathematical definition and descriptive summary of the considered dynamical systems.

Dataset	Method	Time (s)
Exponential Decay	SplineFlow	$1.116$
Exponential Decay	TFM	$0.934$
HopperPhysics	SplineFlow	$5.558$
HopperPhysics	TFM	$2.407$

Appendix

Table 7: Time taken for processing of conditional paths and velocities before training epochs begin

Figure 3: Runtime of adjoint ODE-based and simulation-free flow matching methods.

Figure 4: Runtime comparison between flow matching methods.

Figure 5: Visualizations of the B-spline degree selected via cross-validation vs. the optimal degree, which corresponds to a Spearman correlation coefficient of $\rho=0.85$ .

$\bm{p}$	Model	Exp.	Harm.	Damp.	LV	Hopper	Lorenz
$0$	NeuralODE	$9.99\mathrm{e}{-1}$	$6.54$	$3.47$	$1.98\mathrm{e}{4}$	$2.95$	$1.49$
	LatentODE	$9.80\mathrm{e}{-5}$	$8.20\mathrm{e}{-4}$	$1.21\mathrm{e}{-4}$	$\mathbf{1.56\mathrm{e}{-3}}$	$1.89$	$1.16$
	MMFM	$1.40\mathrm{e}{-3}$	$1.40\mathrm{e}{-2}$	$2.00\mathrm{e}{-3}$	$4.87\mathrm{e}{-1}$	$1.82\mathrm{e}{1}$	$8.02\mathrm{e}{-1}$
	TFM	$6.56\mathrm{e}{-4}$	$1.81\mathrm{e}{-1}$	$1.52\mathrm{e}{-2}$	$1.84\mathrm{e}{-1}$	$1.55$	$2.44$
	SplineFlow (CV)	$\mathbf{8.90\mathrm{e}{-5}}$	$\mathbf{5.28\mathrm{e}{-4}}$	$\mathbf{7.20\mathrm{e}{-5}}$	$5.53\mathrm{e}{-2}$	$\mathbf{1.41}$	$\mathbf{7.77\mathrm{e}{-1}}$
$0.25$	LatentODE	$9.80\mathrm{e}{-5}$	$1.22\mathrm{e}{-3}$	$3.42\mathrm{e}{-4}$	$\mathbf{9.42\mathrm{e}{-4}}$	$1.60$	–

Appendix

Table 8: Comparisons between methods, with SplineFlow degree

m_{\text{CV}}

selected using cross validation.

Setting	$\bm{p}$	SplineFlow (CV)	SplineFlow	SF2M
Exp.	$0$	$3.31\mathrm{e}{-1}$	$3.31\mathrm{e}{-1}$	$\mathbf{3.05\mathrm{e}{-1}}$
Exp.	$0.5$	$\mathbf{1.95\mathrm{e}{-1}}$	$\mathbf{1.95\mathrm{e}{-1}}$	$2.04\mathrm{e}{-1}$
Damp.	$0$	$\mathbf{4.10\mathrm{e}{-1}}$	$\mathbf{4.10\mathrm{e}{-1}}$	$4.30\mathrm{e}{-1}$
Damp.	$0.5$	$3.21\mathrm{e}{-1}$	$3.21\mathrm{e}{-1}$	$\mathbf{3.19\mathrm{e}{-1}}$
LV	$0$	$3.21\mathrm{e}{-1}$	$\mathbf{2.94\mathrm{e}{-1}}$	$4.43\mathrm{e}{-1}$
LV	$0.5$	$3.74\mathrm{e}{-1}$	$\mathbf{3.10\mathrm{e}{-1}}$	$3.75\mathrm{e}{-1}$

Appendix

Table 9:

\mathcal{W}_{2}

comparison of different methods, with SplineFlow degree

m_{\text{CV}}

selected using the CV rule above for SDE systems.

Setting	$\bm{p=0}$	$\bm{p=0.25}$	$\bm{p=0.5}$	$\bm{p=0.75}$
Exp.	$4/4$	$4/4$	$4/4$	$4/5$
Harm.	$3/2$	$3/3$	$3/3$	$5/4$
Damp.	$4/3$	$4/5$	$4/5$	$5/4$
LV	$4/3$	$5/5$	$5/3$	$5/5$
Hopper	$1/1$	$1/1$	$1/1$	$1/1$
Lorenz	$4/3$	–	–	–

Appendix

Table 10:

m_{\mathrm{CV}}

vs.

m_{\mathrm{oracle}}

for ODE and SDE systems. Each entry reports

m_{\mathrm{CV}}/m_{\mathrm{oracle}}

$\bm{p}$	Dataset	$\bm{10^{-4}}$	$\bm{10^{-3}}$	$\bm{10^{-2}}$	$\bm{10^{-1}}$	$\bm{10^{0}}$	$\bm{10^{1}}$
$0$	Exp.	$0.496$	$0.477$	$0.447$	$\mathbf{0.253}$	$6.124$	$116.150$
$0$	Damp.	$0.473$	$0.513$	$\mathbf{0.377}$	$0.402$	$5.478$	$85.405$
$0$	LV	$2.954$	$7.641$	$1.163$	$\mathbf{0.346}$	$0.476$	$0.898$
$0$	Lorenz	$0.201$	$0.210$	$\mathbf{0.185}$	$0.219$	$0.587$	$3.379$
$0.5$	Exp.	$0.426$	$0.382$	$0.392$	$\mathbf{0.142}$	$7.031$	$114.271$
$0.5$	Damp.	$0.499$	$0.490$	$0.502$	$\mathbf{0.278}$	$6.406$	$89.495$

Appendix

Table 11: Ablation over

\sigma

for SDE systems

Panel of Figure 6 — Figure 6: Training loss (log scale) vs. epoch for SplineFlow on ODE systems ( $p=0$ , regular sampling). Shaded bands show the standard deviation across five runs.

$p$	Model	Exp-Decay	Harmonic	Damped Harmonic	Lotka–Volterra	HopperPhysics	Lorenz
$0$	NeuralODE	$0.999\pm 0.126$	$6.542\pm 5.965$	$3.474\pm 0.366$	$1.98\mathrm{e}{4}\pm 1.24\mathrm{e}{4}$	$2.947\pm 0.183$	$1.491\pm 0.359$
	LatentODE	$9.8\mathrm{e}{-5}\pm 4.7\mathrm{e}{-5}$	$8.2\mathrm{e}{-4}\pm 4.2\mathrm{e}{-4}$	$1.2\mathrm{e}{-4}\pm 6.3\mathrm{e}{-4}$	$\mathbf{1.6\mathrm{e}{-3}\pm 1.2\mathrm{e}{-3}}$	$1.894\pm 0.523$	$1.157\pm 0.204$
	MMFM	$1.4\mathrm{e}{-3}\pm 1.0\mathrm{e}{-4}$	$0.014\pm 0.009$	$0.002\pm 0.002$	$0.487\pm 0.384$	$18.179\pm 4.049$	$0.802\pm 0.008$
	TFM	$6.6\mathrm{e}{-4}\pm 8.3\mathrm{e}{-5}$	$0.181\pm 3.8\mathrm{e}{-4}$	$0.015\pm 2.7\mathrm{e}{-4}$	$0.184\pm 0.104$	$1.554\pm 0.297$	$2.442\pm 0.076$
	SplineFlow	$\mathbf{8.9\mathrm{e}{-5}\pm 3.3\mathrm{e}{-5}}$	$\mathbf{3.7\mathrm{e}{-4}\pm 3.1\mathrm{e}{-4}}$	$\mathbf{9.5\mathrm{e}{-5}\pm 1.0\mathrm{e}{-4}}$	$0.086\pm 0.043$	$\mathbf{1.410\pm 0.172}$	$\mathbf{0.639\pm 0.004}$
$0.25$	LatentODE	$9.8\mathrm{e}{-5}\pm 3.7\mathrm{e}{-5}$	$0.001\pm 0.001$	$3.4\mathrm{e}{-4}\pm 5.8\mathrm{e}{-4}$	$\mathbf{9e-4\pm 3.3\mathrm{e}{-4}}$	$1.605\pm 0.019$	–

Appendix

Table 12: Comparisons of different methods for modeling ODE dynamics in MSE metric across

5

deterministic dynamical systems and varying sampling irregularity

p\in\{0,0.25,0.5,0.75\}

Dataset	Model	MSE (P-ODE)	MSE (SDE)	Wasserstein	MMD	Energy
Exp.	SF2M	$3.91\mathrm{e}{-01}\pm 2.26\mathrm{e}{-03}$	$4.77\mathrm{e}{-01}\pm 9.90\mathrm{e}{-03}$	$\mathbf{3.05\mathrm{e}{-01}\pm 4.85\mathrm{e}{-03}}$	$\mathbf{2.40\mathrm{e}{-02}\pm 1.00\mathrm{e}{-03}}$	$\mathbf{6.10\mathrm{e}{-02}\pm 2.62\mathrm{e}{-03}}$
Exp.	SplineFlow	$\mathbf{3.90\mathrm{e}{-01}\pm 4.23\mathrm{e}{-03}}$	$\mathbf{4.55\mathrm{e}{-01}\pm 8.27\mathrm{e}{-03}}$	$3.30\mathrm{e}{-01}\pm 2.83\mathrm{e}{-03}$	$2.70\mathrm{e}{-02}\pm 2.48\mathrm{e}{-03}$	$6.94\mathrm{e}{-02}\pm 1.08\mathrm{e}{-03}$
Damped.	SF2M	$\mathbf{8.44\mathrm{e}{-01}\pm 1.54\mathrm{e}{-02}}$	$\mathbf{9.66\mathrm{e}{-01}\pm 3.29\mathrm{e}{-02}}$	$4.29\mathrm{e}{-01}\pm 7.40\mathrm{e}{-03}$	$4.84\mathrm{e}{-02}\pm 2.96\mathrm{e}{-03}$	$1.25\mathrm{e}{-01}\pm 3.74\mathrm{e}{-03}$
Damped.	SplineFlow	$8.94\mathrm{e}{-01}\pm 5.12\mathrm{e}{-03}$	$9.69\mathrm{e}{-01}\pm 9.41\mathrm{e}{-03}$	$\mathbf{4.10\mathrm{e}{-01}\pm 4.41\mathrm{e}{-03}}$	$\mathbf{4.19\mathrm{e}{-02}\pm 1.54\mathrm{e}{-03}}$	$\mathbf{1.12\mathrm{e}{-01}\pm 7.12\mathrm{e}{-03}}$
LV	SF2M	$4.58\mathrm{e}{-01}\pm 9.74\mathrm{e}{-02}$	$4.60\mathrm{e}{-01}\pm 9.67\mathrm{e}{-02}$	$4.43\mathrm{e}{-01}\pm 4.99\mathrm{e}{-02}$	$1.25\mathrm{e}{-01}\pm 2.35\mathrm{e}{-02}$	$4.65\mathrm{e}{-01}\pm 9.44\mathrm{e}{-02}$
LV	SplineFlow	$\mathbf{2.68\mathrm{e}{-01}\pm 1.44\mathrm{e}{-02}}$	$\mathbf{2.73\mathrm{e}{-01}\pm 1.37\mathrm{e}{-02}}$	$\mathbf{2.93\mathrm{e}{-01}\pm 3.16\mathrm{e}{-03}}$	$\mathbf{4.21\mathrm{e}{-02}\pm 3.45\mathrm{e}{-03}}$	$\mathbf{1.56\mathrm{e}{-01}\pm 1.33\mathrm{e}{-02}}$

Appendix

Table 13: Performance across SDE dynamics datasets for different metrics under regular sampling

Dataset	$\bm{p}$	Model	MSE (P-ODE)	MSE (SDE)	Wasserstein	MMD	Energy
Exp.	$0.25$	SF2M	$\mathbf{3.64\mathrm{e}{-01}\pm 1.27\mathrm{e}{-03}}$	$\mathbf{4.65\mathrm{e}{-01}\pm 6.30\mathrm{e}{-03}}$	$\mathbf{2.45\mathrm{e}{-01}\pm 7.37\mathrm{e}{-03}}$	$\mathbf{1.46\mathrm{e}{-02}\pm 1.15\mathrm{e}{-03}}$	$\mathbf{4.01\mathrm{e}{-02}\pm 2.76\mathrm{e}{-03}}$
	$0.25$	SplineFlow	$3.66\mathrm{e}{-01}\pm 6.08\mathrm{e}{-03}$	$4.69\mathrm{e}{-01}\pm 5.90\mathrm{e}{-03}$	$2.50\mathrm{e}{-01}\pm 5.53\mathrm{e}{-03}$	$1.51\mathrm{e}{-02}\pm 1.03\mathrm{e}{-03}$	$4.12\mathrm{e}{-02}\pm 3.14\mathrm{e}{-03}$
	$0.5$	SF2M	$3.77\mathrm{e}{-01}\pm 5.88\mathrm{e}{-03}$	$\mathbf{5.23\mathrm{e}{-01}\pm 1.07\mathrm{e}{-02}}$	$2.04\mathrm{e}{-01}\pm 1.16\mathrm{e}{-02}$	$9.09\mathrm{e}{-03}\pm 1.93\mathrm{e}{-03}$	$2.83\mathrm{e}{-02}\pm 4.68\mathrm{e}{-03}$
	$0.5$	SplineFlow	$\mathbf{3.76\mathrm{e}{-01}\pm 3.30\mathrm{e}{-03}}$	$5.30\mathrm{e}{-01}\pm 6.59\mathrm{e}{-03}$	$\mathbf{1.94\mathrm{e}{-01}\pm 1.42\mathrm{e}{-03}}$	$\mathbf{7.42\mathrm{e}{-03}\pm 3.31\mathrm{e}{-04}}$	$\mathbf{2.39\mathrm{e}{-02}\pm 1.36\mathrm{e}{-03}}$
	$0.75$	SF2M	$\mathbf{4.05\mathrm{e}{-01}\pm 6.73\mathrm{e}{-03}}$	$\mathbf{6.46\mathrm{e}{-01}\pm 1.07\mathrm{e}{-02}}$	$\mathbf{1.36\mathrm{e}{-01}\pm 1.20\mathrm{e}{-02}}$	$\mathbf{1.47\mathrm{e}{-03}\pm 9.50\mathrm{e}{-04}}$	$\mathbf{1.27\mathrm{e}{-02}\pm 3.13\mathrm{e}{-03}}$
	$0.75$	SplineFlow	$4.18\mathrm{e}{-01}\pm 5.87\mathrm{e}{-03}$	$6.76\mathrm{e}{-01}\pm 8.23\mathrm{e}{-03}$	$1.42\mathrm{e}{-01}\pm 1.72\mathrm{e}{-02}$	$2.05\mathrm{e}{-03}\pm 1.55\mathrm{e}{-03}$	$1.51\mathrm{e}{-02}\pm 5.15\mathrm{e}{-03}$

Appendix

Table 14: Performance across SDE dynamics datasets for different metrics under irregular sampling.

Embedding	Task	Model	Wasserstein	MMD	Energy
PHATE	Interpolation	MOTFM	$7.94e{-1}\!\pm\!2.64e{-2}$	$5.03e{-1}\!\pm\!1.60e{-2}$	$1.44e{+0}\!\pm\!6.21e{-2}$
		SF2M	$7.53e{-1}\!\pm\!2.58e{-2}$	$4.57e{-1}\!\pm\!8.30e{-3}$	$1.31e{+0}\!\pm\!3.55e{-2}$
		SplineFlow	$\mathbf{6.07e{-1}\!\pm\!2.71e{-2}}$	$\mathbf{2.85e{-1}\!\pm\!6.60e{-3}}$	$\mathbf{8.15e{-1}\!\pm\!2.14e{-2}}$
	Extrapolation	MOTFM	$1.57e{+0}\!\pm\!4.50e{-3}$	$1.01e{+0}\!\pm\!1.15e{-2}$	$3.89e{+0}\!\pm\!5.07e{-2}$
		SF2M	$\mathbf{1.53e{+0}\!\pm\!4.72e{-2}}$	$9.72e{-1}\!\pm\!9.20e{-3}$	$3.77e{+0}\!\pm\!7.24e{-2}$
		SplineFlow	$1.59e{+0}\!\pm\!1.92e{-2}$	$\mathbf{9.55e{-1}\!\pm\!8.70e{-3}}$	$\mathbf{3.74e{+0}\!\pm\!4.24e{-2}}$

Appendix

Table 15: Performance of models on holdout interpolation and extrapolation settings for the Brain Regeneration dataset.

Embedding	Task	Model	Wasserstein	MMD	Energy
PHATE	Interpolation	MOTFM	$3.78e{-1}\!\pm\!1.11e{-2}$	$6.01e{-2}\!\pm\!2.52e{-3}$	$1.75e{-1}\!\pm\!4.98e{-3}$
		SF2M	$2.72e{-1}\!\pm\!2.36e{-2}$	$2.46e{-2}\!\pm\!4.03e{-3}$	$8.63e{-2}\!\pm\!1.33e{-2}$
		SplineFlow	$\mathbf{2.07e{-1}\!\pm\!1.69e{-2}}$	$\mathbf{1.26e{-2}\!\pm\!5.87e{-4}}$	$\mathbf{3.96e{-2}\!\pm\!1.40e{-3}}$
	Extrapolation	MOTFM	$4.48e{-1}\!\pm\!5.05e{-2}$	$8.25e{-2}\!\pm\!9.05e{-3}$	$2.39e{-1}\!\pm\!7.18e{-3}$
		SF2M	$\mathbf{4.39e{-1}\!\pm\!4.40e{-2}}$	$\mathbf{5.79e{-2}\!\pm\!1.08e{-2}}$	$\mathbf{1.94e{-1}\!\pm\!1.57e{-2}}$
		SplineFlow	$4.48e{-1}\!\pm\!4.41e{-2}$	$6.90e{-2}\!\pm\!1.70e{-3}$	$2.07e{-1}\!\pm\!2.96e{-2}$

Appendix

Table 16: Performance of models across holdout interpolation and extrapolation settings for the Embryoid Evolution dataset.

Figure 7: Heatmap visualization of relative improvements of LatentODE and SplineFlow in MSE with reference to TFM, summarized from the ODE Table 12 .

Figure 8: TFM evaluated path for Harmonic Oscillator.

Figure 9: SplineFlow evaluated path for Harmonic Oscillator.

Figure 10: LatentODE evaluated path for Harmonic Oscillator.

Figure 11: TFM evaluated path for Damped Harmonic Oscillator.

Figure 12: SplineFlow evaluated path for Damped Harmonic Oscillator.

Figure 13: LatentODE evaluated path for Damped Harmonic Oscillator.

Figure 14: TFM evaluated path for Lotka–Volterra System.

Figure 15: SplineFlow evaluated path for Lotka–Volterra System.

Figure 16: NeuralODE evaluated path for Lorenz System.

Figure 17: LatentODE evaluated path for Lorenz System.

Figure 18: TFM evaluated path for Lorenz System.

Figure 19: TFM evaluated path for Lorenz System.

Figure 20: SplineFlow evaluated path for Lorenz System.

Figure 21: SplineFlow evaluated path for Lorenz System.

Figure 22: SF2M evaluated path for Lorenz System.

Figure 23: SF2M evaluated path for Lorenz System.

Figure 24: SplineFlow evaluated path for Lorenz System.

Figure 25: SplineFlow evaluated path for Lorenz System.

Degree	$\bm{p=0}$	$\bm{p=0.25}$	$\bm{p=0.5}$	$\bm{p=0.75}$
1	$1.97e{-2}\!\pm\!4.83e{-4}$	$2.05e{-2}\!\pm\!2.69e{-3}$	$3.19e{-2}\!\pm\!4.18e{-3}$	$1.71e{-1}\!\pm\!3.71e{-2}$
2	$6.5e{-5}\!\pm\!3.47e{-5}$	$6.2e{-5}\!\pm\!2.69e{-5}$	$5.4e{-5}\!\pm\!3.44e{-5}$	$4.15e{-4}\!\pm\!8.13e{-5}$
3	$3.1e{-5}\!\pm\!8.47e{-6}$	$3.4e{-5}\!\pm\!9.69e{-6}$	$1.00e{-4}\!\pm\!8.68e{-5}$	$1.84e{-4}\!\pm\!4.29e{-5}$
4	$7.2e{-5}\!\pm\!3.02e{-5}$	$5.1e{-5}\!\pm\!3.44e{-5}$	$4.7e{-5}\!\pm\!3.37e{-5}$	$3.4e{-5}\!\pm\!4.54e{-6}$
5	$9.5e{-5}\!\pm\!1.02e{-4}$	$3.0e{-5}\!\pm\!4.77e{-6}$	$4.7e{-5}\!\pm\!2.14e{-5}$	$4.8e{-5}\!\pm\!2.30e{-5}$

Appendix

Table 17: Effect of spline degree on MSE performance of SplineFlow on damped harmonic system across missingness probabilities

p

Degree	$\bm{p=0}$	$\bm{p=0.25}$	$\bm{p=0.5}$	$\bm{p=0.75}$
1	$4.19e{+0}\!\pm\!7.51e{+0}$	$4.29e{+0}\!\pm\!6.00e{+0}$	$1.22e{+1}\!\pm\!8.84e{+0}$	$2.15e{+2}\!\pm\!1.28e{+2}$
2	$5.83e{-2}\!\pm\!4.14e{-2}$	$2.56e{-1}\!\pm\!2.05e{-1}$	$3.56e{-1}\!\pm\!2.24e{-1}$	$6.82e{+0}\!\pm\!3.98e{+0}$
3	$4.50e{-2}\!\pm\!2.78e{-2}$	$1.87e{-1}\!\pm\!1.33e{-1}$	$2.75e{-1}\!\pm\!1.89e{-1}$	$8.88e{+0}\!\pm\!8.82e{+0}$
4	$5.53e{-2}\!\pm\!2.36e{-2}$	$1.83e{-1}\!\pm\!1.41e{-1}$	$4.20e{+0}\!\pm\!7.96e{+0}$	$4.31e{+0}\!\pm\!4.40e{+0}$
5	$8.55e{-2}\!\pm\!4.28e{-2}$	$1.00e{-1}\!\pm\!5.72e{-2}$	$7.05e{-1}\!\pm\!1.21e{+0}$	$1.78e{+0}\!\pm\!1.43e{+0}$

Appendix

Table 18: Effect of spline degree on MSE performance of SplineFlow on Lotka–Volterra system across missingness probabilities

p

Degree	$\bm{p=0}$	$\bm{p=0.25}$	$\bm{p=0.5}$	$\bm{p=0.75}$
1	$1.41e{+0}\!\pm\!1.72e{-1}$	$1.43e{+0}\!\pm\!2.98e{-2}$	$1.83e{+0}\!\pm\!6.19e{-2}$	$3.34e{+0}\!\pm\!4.30e{-1}$
2	$6.60e{+0}\!\pm\!2.35e{+0}$	$5.39e{+1}\!\pm\!2.47e{+1}$	$2.80e{+2}\!\pm\!5.93e{+1}$	$1.00e{+3}\!\pm\!6.07e{+1}$
3	$1.25e{+1}\!\pm\!8.86e{+0}$	$3.91e{+2}\!\pm\!1.80e{+2}$	$6.38e{+4}\!\pm\!1.24e{+4}$	$1.73e{+4}\!\pm\!1.56e{+3}$
4	$1.63e{+1}\!\pm\!1.29e{+1}$	$1.41e{+5}\!\pm\!2.11e{+4}$	$4.64e{+5}\!\pm\!1.50e{+5}$	$1.20e{+5}\!\pm\!6.57e{+4}$
5	$2.26e{+1}\!\pm\!6.33e{+0}$	$8.95e{+5}\!\pm\!1.36e{+5}$	$6.71e{+5}\!\pm\!1.30e{+5}$	$9.25e{+4}\!\pm\!2.44e{+4}$

Appendix

Table 19: Effect of spline degree on MSE performance of SplineFlow on hopperphysics simulations across missingness probabilities

p

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$3.91e{-1}\!\pm\!4.23e{-3}$	$4.56e{-1}\!\pm\!8.27e{-3}$	$3.31e{-1}\!\pm\!2.83e{-3}$	$2.73e{-2}\!\pm\!2.48e{-3}$	$6.94e{-2}\!\pm\!1.08e{-3}$
2	$1.25e{+0}\!\pm\!4.04e{-1}$	$2.38e{+0}\!\pm\!5.77e{-1}$	$7.94e{-1}\!\pm\!1.78e{-1}$	$3.38e{-2}\!\pm\!3.08e{-3}$	$1.12e{-1}\!\pm\!1.67e{-2}$
3	$8.22e{-1}\!\pm\!6.78e{-2}$	$2.03e{+0}\!\pm\!4.47e{-1}$	$6.87e{-1}\!\pm\!1.70e{-1}$	$3.85e{-2}\!\pm\!1.01e{-2}$	$1.25e{-1}\!\pm\!3.65e{-2}$
4	$1.20e{+0}\!\pm\!6.28e{-1}$	$4.74e{+0}\!\pm\!4.43e{+0}$	$1.17e{+0}\!\pm\!8.69e{-1}$	$3.46e{-2}\!\pm\!1.40e{-2}$	$1.30e{-1}\!\pm\!7.66e{-2}$
5	$1.27e{+0}\!\pm\!3.89e{-1}$	$9.89e{+0}\!\pm\!1.17e{+1}$	$1.49e{+0}\!\pm\!1.31e{+0}$	$6.13e{-2}\!\pm\!1.22e{-2}$	$2.26e{-1}\!\pm\!9.57e{-2}$

Appendix

Table 20: Effect of spline order on Exp-Decay performance.

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$3.66e{-1}\!\pm\!6.08e{-3}$	$4.70e{-1}\!\pm\!5.90e{-3}$	$2.51e{-1}\!\pm\!5.53e{-3}$	$1.51e{-2}\!\pm\!1.03e{-3}$	$4.12e{-2}\!\pm\!3.14e{-3}$
2	$7.30e{-1}\!\pm\!1.41e{-1}$	$1.49e{+0}\!\pm\!3.84e{-2}$	$5.52e{-1}\!\pm\!3.67e{-2}$	$4.28e{-2}\!\pm\!1.87e{-2}$	$1.43e{-1}\!\pm\!6.38e{-2}$
3	$1.37e{+0}\!\pm\!8.59e{-1}$	$2.55e{+0}\!\pm\!9.96e{-1}$	$7.75e{-1}\!\pm\!2.18e{-1}$	$4.36e{-2}\!\pm\!1.19e{-2}$	$1.47e{-1}\!\pm\!4.29e{-2}$
4	$8.48e{-1}\!\pm\!2.68e{-1}$	$2.73e{+0}\!\pm\!1.64e{+0}$	$8.58e{-1}\!\pm\!3.85e{-1}$	$4.18e{-2}\!\pm\!7.33e{-3}$	$1.38e{-1}\!\pm\!3.88e{-2}$
5	$3.12e{+0}\!\pm\!1.10e{+0}$	$4.41e{+0}\!\pm\!8.47e{-1}$	$1.17e{+0}\!\pm\!1.97e{-1}$	$6.01e{-2}\!\pm\!1.21e{-2}$	$2.60e{-1}\!\pm\!4.65e{-2}$

Appendix

Table 21: Effect of B-spline degree on Exp-decay performance for

p=0.25

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$3.76e{-1}\!\pm\!3.30e{-3}$	$5.30e{-1}\!\pm\!6.59e{-3}$	$1.95e{-1}\!\pm\!1.42e{-3}$	$7.42e{-3}\!\pm\!3.31e{-4}$	$2.40e{-2}\!\pm\!1.36e{-3}$
2	$6.44e{-1}\!\pm\!5.12e{-2}$	$1.80e{+0}\!\pm\!5.27e{-1}$	$5.92e{-1}\!\pm\!1.49e{-1}$	$2.92e{-2}\!\pm\!8.18e{-3}$	$8.90e{-2}\!\pm\!1.62e{-2}$
3	$6.52e{-1}\!\pm\!9.32e{-2}$	$1.61e{+0}\!\pm\!9.35e{-2}$	$5.57e{-1}\!\pm\!4.34e{-2}$	$2.34e{-2}\!\pm\!9.21e{-3}$	$7.47e{-2}\!\pm\!2.55e{-2}$
4	$8.61e{-1}\!\pm\!3.17e{-1}$	$2.78e{+0}\!\pm\!1.86e{+0}$	$8.49e{-1}\!\pm\!4.06e{-1}$	$3.21e{-2}\!\pm\!1.25e{-2}$	$1.34e{-1}\!\pm\!5.61e{-2}$
5	$2.75e{+0}\!\pm\!2.96e{+0}$	$3.53e{+0}\!\pm\!3.21e{+0}$	$9.04e{-1}\!\pm\!6.04e{-1}$	$7.77e{-2}\!\pm\!5.26e{-2}$	$3.93e{-1}\!\pm\!3.49e{-1}$

Appendix

Table 22: Effect of B-spline degree on Exp-decay performance for

p=0.5

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$4.19e{-1}\!\pm\!5.87e{-3}$	$6.77e{-1}\!\pm\!8.23e{-3}$	$1.42e{-1}\!\pm\!1.72e{-2}$	$2.06e{-3}\!\pm\!1.55e{-3}$	$1.51e{-2}\!\pm\!5.15e{-3}$
2	$7.83e{-1}\!\pm\!4.41e{-2}$	$2.24e{+0}\!\pm\!1.00e{+0}$	$7.53e{-1}\!\pm\!2.53e{-1}$	$3.91e{-2}\!\pm\!9.25e{-3}$	$1.43e{-1}\!\pm\!5.00e{-2}$
3	$9.82e{-1}\!\pm\!7.21e{-2}$	$5.56e{+0}\!\pm\!3.84e{+0}$	$1.39e{+0}\!\pm\!6.96e{-1}$	$4.30e{-2}\!\pm\!1.32e{-2}$	$1.69e{-1}\!\pm\!3.47e{-2}$
4	$1.12e{+0}\!\pm\!9.37e{-1}$	$2.61e{+0}\!\pm\!5.93e{-1}$	$9.23e{-1}\!\pm\!1.44e{-1}$	$5.42e{-2}\!\pm\!2.47e{-2}$	$2.26e{-1}\!\pm\!1.46e{-1}$
5	$3.18e{+5}\!\pm\!4.50e{+5}$	$3.03e{+5}\!\pm\!4.23e{+5}$	$1.99e{+2}\!\pm\!2.47e{+2}$	$2.38e{-1}\!\pm\!2.83e{-1}$	$2.61e{+2}\!\pm\!3.68e{+2}$

Appendix

Table 23: Effect of B-spline degree on Exp-decay performance for

p=0.75

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$4.70e{-1}\!\pm\!9.54e{-2}$	$4.69e{-1}\!\pm\!9.23e{-2}$	$4.27e{-1}\!\pm\!7.98e{-2}$	$1.12e{-1}\!\pm\!6.87e{-2}$	$4.26e{-1}\!\pm\!2.38e{-1}$
2	$2.68e{-1}\!\pm\!1.44e{-2}$	$2.73e{-1}\!\pm\!1.37e{-2}$	$2.94e{-1}\!\pm\!3.16e{-3}$	$4.21e{-2}\!\pm\!3.45e{-3}$	$1.56e{-1}\!\pm\!1.33e{-2}$
3	$3.03e{-1}\!\pm\!6.56e{-3}$	$3.03e{-1}\!\pm\!1.10e{-2}$	$3.21e{-1}\!\pm\!8.22e{-3}$	$4.64e{-2}\!\pm\!3.21e{-3}$	$1.74e{-1}\!\pm\!1.36e{-2}$
4	$3.47e{-1}\!\pm\!3.19e{-2}$	$3.47e{-1}\!\pm\!3.10e{-2}$	$3.60e{-1}\!\pm\!2.25e{-2}$	$6.97e{-2}\!\pm\!1.63e{-2}$	$2.60e{-1}\!\pm\!5.59e{-2}$
5	$3.28e{-1}\!\pm\!2.62e{-2}$	$3.30e{-1}\!\pm\!2.62e{-2}$	$3.50e{-1}\!\pm\!3.36e{-2}$	$6.87e{-2}\!\pm\!2.85e{-2}$	$2.52e{-1}\!\pm\!9.45e{-2}$

Appendix

Table 24: Effect of spline order on Lotka–Volterra performance.

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$5.22e{-1}\!\pm\!1.72e{-1}$	$5.24e{-1}\!\pm\!1.71e{-1}$	$4.56e{-1}\!\pm\!1.33e{-1}$	$1.32e{-1}\!\pm\!1.11e{-1}$	$4.89e{-1}\!\pm\!3.77e{-1}$
2	$3.40e{-1}\!\pm\!5.04e{-2}$	$3.38e{-1}\!\pm\!4.76e{-2}$	$3.20e{-1}\!\pm\!1.01e{-2}$	$4.57e{-2}\!\pm\!2.87e{-3}$	$1.82e{-1}\!\pm\!1.18e{-2}$
3	$3.34e{-1}\!\pm\!5.05e{-2}$	$3.39e{-1}\!\pm\!4.86e{-2}$	$3.37e{-1}\!\pm\!1.39e{-2}$	$6.22e{-2}\!\pm\!1.12e{-2}$	$2.47e{-1}\!\pm\!5.83e{-2}$
4	$3.18e{-1}\!\pm\!3.58e{-2}$	$3.21e{-1}\!\pm\!3.21e{-2}$	$3.19e{-1}\!\pm\!1.21e{-2}$	$4.45e{-2}\!\pm\!5.03e{-3}$	$1.75e{-1}\!\pm\!2.60e{-2}$
5	$2.81e{-1}\!\pm\!1.70e{-2}$	$2.83e{-1}\!\pm\!1.54e{-2}$	$3.19e{-1}\!\pm\!5.33e{-3}$	$4.83e{-2}\!\pm\!3.97e{-3}$	$1.83e{-1}\!\pm\!1.57e{-2}$

Appendix

Table 25: Effect of B-spline degree on Lotka–Volterra performance for

p=0.25

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$3.82e{-1}\!\pm\!6.10e{-2}$	$3.83e{-1}\!\pm\!5.94e{-2}$	$3.70e{-1}\!\pm\!1.68e{-2}$	$7.09e{-2}\!\pm\!4.52e{-3}$	$2.75e{-1}\!\pm\!1.38e{-2}$
2	$4.11e{-1}\!\pm\!4.68e{-2}$	$4.15e{-1}\!\pm\!4.79e{-2}$	$3.74e{-1}\!\pm\!3.81e{-2}$	$6.24e{-2}\!\pm\!9.27e{-3}$	$2.59e{-1}\!\pm\!3.66e{-2}$
3	$2.88e{-1}\!\pm\!2.32e{-2}$	$2.85e{-1}\!\pm\!2.54e{-2}$	$3.10e{-1}\!\pm\!1.16e{-2}$	$4.42e{-2}\!\pm\!3.80e{-3}$	$1.64e{-1}\!\pm\!2.32e{-2}$
4	$3.70e{-1}\!\pm\!1.10e{-1}$	$3.72e{-1}\!\pm\!1.11e{-1}$	$3.48e{-1}\!\pm\!4.99e{-2}$	$5.39e{-2}\!\pm\!2.09e{-2}$	$2.22e{-1}\!\pm\!1.02e{-1}$
5	$3.19e{-1}\!\pm\!2.96e{-2}$	$3.23e{-1}\!\pm\!3.05e{-2}$	$3.11e{-1}\!\pm\!1.57e{-2}$	$4.50e{-2}\!\pm\!2.94e{-3}$	$1.79e{-1}\!\pm\!7.45e{-3}$

Appendix

Table 26: Effect of B-spline degree on Lotka–Volterra performance for

p=0.5

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$4.98e{-1}\!\pm\!1.36e{-1}$	$4.99e{-1}\!\pm\!1.35e{-1}$	$4.53e{-1}\!\pm\!4.31e{-2}$	$1.21e{-1}\!\pm\!1.03e{-2}$	$4.53e{-1}\!\pm\!7.16e{-2}$
2	$4.30e{-1}\!\pm\!7.67e{-2}$	$4.33e{-1}\!\pm\!7.35e{-2}$	$3.97e{-1}\!\pm\!6.21e{-2}$	$8.44e{-2}\!\pm\!3.38e{-2}$	$3.30e{-1}\!\pm\!1.15e{-1}$
3	$3.77e{-1}\!\pm\!5.73e{-2}$	$3.78e{-1}\!\pm\!5.48e{-2}$	$3.61e{-1}\!\pm\!4.04e{-2}$	$5.57e{-2}\!\pm\!9.43e{-3}$	$2.23e{-1}\!\pm\!4.77e{-2}$
4	$4.49e{-1}\!\pm\!1.40e{-1}$	$4.43e{-1}\!\pm\!1.29e{-1}$	$3.55e{-1}\!\pm\!4.84e{-2}$	$6.56e{-2}\!\pm\!1.72e{-2}$	$2.75e{-1}\!\pm\!9.15e{-2}$
5	$6.03e{-1}\!\pm\!2.33e{-1}$	$5.98e{-1}\!\pm\!2.31e{-1}$	$4.60e{-1}\!\pm\!7.29e{-2}$	$1.26e{-1}\!\pm\!4.30e{-2}$	$5.06e{-1}\!\pm\!1.91e{-1}$

Appendix

Table 27: Effect of B-spline degree on Lotka–Volterra performance for

p=0.75

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$8.95e{-1}\!\pm\!5.12e{-3}$	$9.70e{-1}\!\pm\!9.41e{-3}$	$4.10e{-1}\!\pm\!4.41e{-3}$	$4.19e{-2}\!\pm\!1.54e{-3}$	$1.12e{-1}\!\pm\!7.12e{-3}$
2	$1.78e{+0}\!\pm\!1.96e{-1}$	$8.02e{+0}\!\pm\!4.45e{+0}$	$1.58e{+0}\!\pm\!6.03e{-1}$	$7.48e{-3}\!\pm\!1.30e{-3}$	$8.23e{-2}\!\pm\!3.24e{-2}$
3	$5.51e{+0}\!\pm\!2.89e{+0}$	$1.30e{+1}\!\pm\!7.52e{+0}$	$2.07e{+0}\!\pm\!6.98e{-1}$	$2.08e{-2}\!\pm\!1.41e{-2}$	$2.06e{-1}\!\pm\!1.72e{-1}$
4	$9.23e{+0}\!\pm\!2.11e{+0}$	$1.56e{+1}\!\pm\!2.84e{+0}$	$2.43e{+0}\!\pm\!2.60e{-1}$	$1.69e{-2}\!\pm\!7.13e{-3}$	$1.90e{-1}\!\pm\!4.45e{-2}$
5	$8.53e{+0}\!\pm\!5.52e{+0}$	$2.21e{+1}\!\pm\!1.21e{+1}$	$3.04e{+0}\!\pm\!7.89e{-1}$	$2.13e{-2}\!\pm\!1.77e{-3}$	$2.67e{-1}\!\pm\!1.09e{-1}$

Appendix

Table 28: Effect of spline order on damped harmonic oscillator performance.

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$7.99e{-1}\!\pm\!1.57e{-2}$	$9.14e{-1}\!\pm\!2.28e{-2}$	$3.65e{-1}\!\pm\!2.11e{-2}$	$3.42e{-2}\!\pm\!4.34e{-3}$	$9.17e{-2}\!\pm\!1.07e{-2}$
2	$2.22e{+0}\!\pm\!8.18e{-1}$	$3.48e{+0}\!\pm\!7.77e{-1}$	$8.66e{-1}\!\pm\!1.83e{-1}$	$1.43e{-2}\!\pm\!2.24e{-3}$	$9.77e{-2}\!\pm\!7.13e{-3}$
3	$1.49e{+0}\!\pm\!3.66e{-2}$	$6.24e{+0}\!\pm\!2.46e{+0}$	$1.33e{+0}\!\pm\!4.36e{-1}$	$7.18e{-3}\!\pm\!3.63e{-3}$	$7.58e{-2}\!\pm\!2.36e{-2}$
4	$4.11e{+0}\!\pm\!8.82e{-1}$	$9.91e{+0}\!\pm\!3.36e{+0}$	$1.76e{+0}\!\pm\!2.72e{-1}$	$2.62e{-2}\!\pm\!2.77e{-3}$	$1.96e{-1}\!\pm\!2.59e{-2}$
5	$7.85e{+0}\!\pm\!3.86e{+0}$	$1.52e{+1}\!\pm\!7.33e{+0}$	$2.34e{+0}\!\pm\!7.83e{-1}$	$3.32e{-2}\!\pm\!2.33e{-2}$	$3.66e{-1}\!\pm\!1.77e{-1}$

Appendix

Table 29: Effect of B-spline degree on Damped-Harmonic performance for

p=0.25

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$8.05e{-1}\!\pm\!8.73e{-3}$	$9.81e{-1}\!\pm\!3.99e{-2}$	$3.21e{-1}\!\pm\!2.84e{-2}$	$2.18e{-2}\!\pm\!5.59e{-3}$	$6.87e{-2}\!\pm\!1.21e{-2}$
2	$1.61e{+0}\!\pm\!4.11e{-1}$	$9.25e{+0}\!\pm\!7.31e{+0}$	$1.52e{+0}\!\pm\!9.81e{-1}$	$1.23e{-2}\!\pm\!7.06e{-3}$	$1.04e{-1}\!\pm\!6.55e{-2}$
3	$1.34e{+0}\!\pm\!2.35e{-1}$	$3.23e{+0}\!\pm\!1.62e{+0}$	$7.52e{-1}\!\pm\!3.50e{-1}$	$1.05e{-2}\!\pm\!4.37e{-3}$	$7.49e{-2}\!\pm\!2.28e{-2}$
4	$1.64e{+0}\!\pm\!4.51e{-1}$	$1.38e{+1}\!\pm\!1.04e{+1}$	$2.08e{+0}\!\pm\!1.07e{+0}$	$1.02e{-2}\!\pm\!3.90e{-3}$	$9.98e{-2}\!\pm\!4.27e{-2}$
5	$4.26e{+0}\!\pm\!2.48e{+0}$	$2.05e{+1}\!\pm\!2.01e{+1}$	$2.71e{+0}\!\pm\!1.93e{+0}$	$5.87e{-3}\!\pm\!2.05e{-3}$	$6.62e{-2}\!\pm\!2.93e{-2}$

Appendix

Table 30: Effect of B-spline degree on Damped-Harmonic performance for

p=0.5

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$9.38e{-1}\!\pm\!2.89e{-2}$	$1.17e{+0}\!\pm\!1.17e{-2}$	$2.86e{-1}\!\pm\!1.10e{-2}$	$1.24e{-2}\!\pm\!7.65e{-4}$	$5.28e{-2}\!\pm\!4.89e{-3}$
2	$2.24e{+0}\!\pm\!6.32e{-1}$	$7.16e{+0}\!\pm\!5.03e{-1}$	$1.56e{+0}\!\pm\!1.18e{-1}$	$1.29e{-2}\!\pm\!4.25e{-3}$	$1.02e{-1}\!\pm\!3.60e{-2}$
3	$3.34e{+0}\!\pm\!2.64e{+0}$	$8.62e{+0}\!\pm\!5.79e{+0}$	$1.43e{+0}\!\pm\!5.99e{-1}$	$1.13e{-2}\!\pm\!1.98e{-3}$	$9.04e{-2}\!\pm\!1.63e{-2}$
4	$7.48e{+0}\!\pm\!8.12e{+0}$	$3.67e{+1}\!\pm\!2.28e{+1}$	$3.83e{+0}\!\pm\!1.39e{+0}$	$3.72e{-2}\!\pm\!3.14e{-2}$	$2.98e{-1}\!\pm\!2.59e{-1}$
5	$8.79e{+1}\!\pm\!1.22e{+2}$	$4.87e{+1}\!\pm\!5.88e{+1}$	$3.11e{+0}\!\pm\!2.19e{+0}$	$3.30e{-2}\!\pm\!3.02e{-2}$	$3.06e{-1}\!\pm\!3.25e{-1}$

Appendix

Table 31: Effect of B-spline degree on Damped-Harmonic performance for

p=0.75

Degree	MSE
1	$1.380\pm 1.890\mathrm{e}{-02}$
2	$6.800\mathrm{e}{-01}\pm 1.520\mathrm{e}{-02}$
3	$6.400\mathrm{e}{-01}\pm 3.790\mathrm{e}{-03}$
4	$7.800\mathrm{e}{-01}\pm 1.320\mathrm{e}{-01}$
5	$9.200\mathrm{e}{-01}\pm 7.360\mathrm{e}{-02}$

Appendix

Table 32: Lorenz system results for regularly sampled observations

p=0

Degree	MSE ${}_{\text{PODE}}$	MSE ${}_{\text{SDE}}$	Wasserstein	MMD	Energy
1	$2.21e{+0}\!\pm\!9.71e{-3}$	$2.21e{+0}\!\pm\!9.67e{-3}$	$6.31e{-1}\!\pm\!2.03e{-3}$	$1.30e{-1}\!\pm\!1.22e{-3}$	$4.01e{-1}\!\pm\!3.50e{-3}$
2	$2.19e{+0}\!\pm\!1.38e{-2}$	$2.19e{+0}\!\pm\!1.88e{-2}$	$6.30e{-1}\!\pm\!1.90e{-3}$	$1.29e{-1}\!\pm\!2.21e{-4}$	$3.98e{-1}\!\pm\!1.52e{-3}$
3	$4.82e{+0}\!\pm\!4.96e{+0}$	$3.71e{+0}\!\pm\!3.37e{+0}$	$9.31e{-1}\!\pm\!9.85e{-1}$	$7.44e{-3}\!\pm\!6.53e{-3}$	$5.61e{-2}\!\pm\!3.44e{-2}$
4	$1.23e{+0}\!\pm\!4.01e{-2}$	$1.23e{+0}\!\pm\!5.62e{-2}$	$1.80e{-1}\!\pm\!1.47e{-2}$	$4.51e{-3}\!\pm\!1.79e{-3}$	$3.66e{-2}\!\pm\!7.65e{-3}$
5	$1.36e{+0}\!\pm\!6.41e{-2}$	$1.50e{+0}\!\pm\!2.46e{-1}$	$3.07e{-1}\!\pm\!1.49e{-1}$	$6.44e{-3}\!\pm\!1.73e{-3}$	$4.47e{-2}\!\pm\!1.08e{-2}$

Appendix

Table 33: Effect of spline order on stochastic Lorenz system performance.

Degree	Wasserstein	MMD	Energy
1	$7.714\mathrm{e}{-01}\pm 1.250\mathrm{e}{-02}$	$4.813\mathrm{e}{-01}\pm 1.740\mathrm{e}{-02}$	$1.391\mathrm{e}{+00}\pm 7.250\mathrm{e}{-02}$
2	$6.421\mathrm{e}{-01}\pm 5.280\mathrm{e}{-02}$	$3.088\mathrm{e}{-01}\pm 2.570\mathrm{e}{-02}$	$9.172\mathrm{e}{-01}\pm 9.580\mathrm{e}{-02}$
3	$6.519\mathrm{e}{-01}\pm 2.100\mathrm{e}{-02}$	$3.174\mathrm{e}{-01}\pm 5.200\mathrm{e}{-03}$	$9.277\mathrm{e}{-01}\pm 2.030\mathrm{e}{-02}$
4	$6.548\mathrm{e}{-01}\pm 2.930\mathrm{e}{-02}$	$3.120\mathrm{e}{-01}\pm 2.020\mathrm{e}{-02}$	$9.189\mathrm{e}{-01}\pm 7.080\mathrm{e}{-02}$
5	$6.066\mathrm{e}{-01}\pm 2.710\mathrm{e}{-02}$	$2.846\mathrm{e}{-01}\pm 6.600\mathrm{e}{-03}$	$8.152\mathrm{e}{-01}\pm 2.140\mathrm{e}{-02}$

Appendix

Table 34: Effect of B-spline degree on PHATE interpolation performance.

Degree	Wasserstein	MMD	Energy
1	$1.588\pm 2.270\mathrm{e}{-02}$	$9.796\mathrm{e}{-01}\pm 4.000\mathrm{e}{-03}$	$3.843\pm 3.700\mathrm{e}{-02}$
2	$1.584\pm 3.670\mathrm{e}{-02}$	$9.654\mathrm{e}{-01}\pm 6.300\mathrm{e}{-03}$	$3.787\pm 6.170\mathrm{e}{-02}$
3	$1.587\pm 1.700\mathrm{e}{-02}$	$9.623\mathrm{e}{-01}\pm 2.700\mathrm{e}{-03}$	$3.761\pm 2.500\mathrm{e}{-02}$
4	$1.536\pm 1.920\mathrm{e}{-02}$	$9.550\mathrm{e}{-01}\pm 8.700\mathrm{e}{-03}$	$3.740\pm 4.240\mathrm{e}{-02}$
5	$1.545\pm 2.720\mathrm{e}{-02}$	$9.644\mathrm{e}{-01}\pm 3.200\mathrm{e}{-03}$	$3.751\pm 1.290\mathrm{e}{-02}$

Appendix

Table 35: Effect of B-spline degree on PHATE extrapolation performance.

Degree	Wasserstein	MMD	Energy
1	$8.719\mathrm{e}{-01}\pm 3.180\mathrm{e}{-02}$	$6.320\mathrm{e}{-02}\pm 3.600\mathrm{e}{-03}$	$1.120\pm 4.510\mathrm{e}{-02}$
2	$8.380\mathrm{e}{-01}\pm 1.370\mathrm{e}{-02}$	$5.820\mathrm{e}{-02}\pm 2.500\mathrm{e}{-03}$	$1.019\pm 3.930\mathrm{e}{-02}$
3	$8.346\mathrm{e}{-01}\pm 1.670\mathrm{e}{-02}$	$6.030\mathrm{e}{-02}\pm 1.200\mathrm{e}{-03}$	$1.035\pm 1.660\mathrm{e}{-02}$
4	$8.502\mathrm{e}{-01}\pm 1.110\mathrm{e}{-02}$	$6.140\mathrm{e}{-02}\pm 3.000\mathrm{e}{-03}$	$1.063\pm 3.660\mathrm{e}{-02}$
5	$8.512\mathrm{e}{-01}\pm 1.940\mathrm{e}{-02}$	$6.150\mathrm{e}{-02}\pm 2.600\mathrm{e}{-03}$	$1.063\pm 3.940\mathrm{e}{-02}$

Appendix

Table 36: Effect of B-spline degree on PCA interpolation performance.

Degree	Wasserstein	MMD	Energy
1	$8.434\mathrm{e}{-01}\pm 2.500\mathrm{e}{-02}$	$1.730\mathrm{e}{-01}\pm 8.800\mathrm{e}{-03}$	$1.869\pm 2.100\mathrm{e}{-02}$
2	$8.040\mathrm{e}{-01}\pm 3.720\mathrm{e}{-02}$	$1.292\mathrm{e}{-01}\pm 6.800\mathrm{e}{-03}$	$1.645\pm 5.660\mathrm{e}{-02}$
3	$8.622\mathrm{e}{-01}\pm 4.030\mathrm{e}{-02}$	$1.415\mathrm{e}{-01}\pm 7.900\mathrm{e}{-03}$	$1.738\pm 7.230\mathrm{e}{-02}$
4	$8.206\mathrm{e}{-01}\pm 4.100\mathrm{e}{-02}$	$1.355\mathrm{e}{-01}\pm 1.180\mathrm{e}{-02}$	$1.736\pm 8.500\mathrm{e}{-02}$
5	$8.234\mathrm{e}{-01}\pm 1.070\mathrm{e}{-02}$	$1.336\mathrm{e}{-01}\pm 1.310\mathrm{e}{-02}$	$1.690\pm 1.068\mathrm{e}{-01}$

Appendix

Table 37: Effect of B-spline degree on PCA extrapolation performance.

Degree	Wasserstein	MMD	Energy
1	$2.355\mathrm{e}{-01}\pm 3.280\mathrm{e}{-02}$	$2.783\mathrm{e}{-02}\pm 5.360\mathrm{e}{-03}$	$7.954\mathrm{e}{-02}\pm 1.660\mathrm{e}{-02}$
2	$2.745\mathrm{e}{-01}\pm 3.700\mathrm{e}{-02}$	$1.768\mathrm{e}{-02}\pm 3.220\mathrm{e}{-03}$	$6.331\mathrm{e}{-02}\pm 1.540\mathrm{e}{-02}$
3	$2.071\mathrm{e}{-01}\pm 1.690\mathrm{e}{-02}$	$1.260\mathrm{e}{-02}\pm 5.870\mathrm{e}{-04}$	$3.963\mathrm{e}{-02}\pm 1.400\mathrm{e}{-03}$
4	$2.560\mathrm{e}{-01}\pm 6.300\mathrm{e}{-02}$	$1.592\mathrm{e}{-02}\pm 4.450\mathrm{e}{-03}$	$5.462\mathrm{e}{-02}\pm 1.620\mathrm{e}{-02}$
5	$2.217\mathrm{e}{-01}\pm 3.530\mathrm{e}{-02}$	$1.408\mathrm{e}{-02}\pm 3.100\mathrm{e}{-03}$	$4.467\mathrm{e}{-02}\pm 9.410\mathrm{e}{-03}$

Appendix

Table 38: Effect of B-spline degree on PHATE interpolation performance.

Degree	Wasserstein	MMD	Energy
1	$5.765\mathrm{e}{-01}\pm 5.860\mathrm{e}{-02}$	$7.572\mathrm{e}{-02}\pm 1.110\mathrm{e}{-02}$	$2.833\mathrm{e}{-01}\pm 4.370\mathrm{e}{-02}$
2	$4.483\mathrm{e}{-01}\pm 4.410\mathrm{e}{-02}$	$6.902\mathrm{e}{-02}\pm 1.700\mathrm{e}{-03}$	$2.073\mathrm{e}{-01}\pm 2.960\mathrm{e}{-02}$
3	$4.774\mathrm{e}{-01}\pm 6.720\mathrm{e}{-02}$	$6.540\mathrm{e}{-02}\pm 9.270\mathrm{e}{-03}$	$2.367\mathrm{e}{-01}\pm 5.200\mathrm{e}{-02}$
4	$5.502\mathrm{e}{-01}\pm 7.380\mathrm{e}{-02}$	$7.318\mathrm{e}{-02}\pm 1.650\mathrm{e}{-02}$	$2.916\mathrm{e}{-01}\pm 8.900\mathrm{e}{-02}$
5	$4.733\mathrm{e}{-01}\pm 6.450\mathrm{e}{-02}$	$7.417\mathrm{e}{-02}\pm 3.690\mathrm{e}{-03}$	$2.423\mathrm{e}{-01}\pm 2.410\mathrm{e}{-02}$

Appendix

Table 39: Effect of B-spline degree on PHATE extrapolation performance.

Degree	Wasserstein	MMD	Energy
1	$3.008\mathrm{e}{-01}\pm 6.130\mathrm{e}{-03}$	$1.341\mathrm{e}{-02}\pm 7.060\mathrm{e}{-04}$	$1.913\mathrm{e}{-01}\pm 8.150\mathrm{e}{-03}$
2	$3.073\mathrm{e}{-01}\pm 1.260\mathrm{e}{-02}$	$1.231\mathrm{e}{-02}\pm 5.310\mathrm{e}{-04}$	$2.070\mathrm{e}{-01}\pm 9.860\mathrm{e}{-03}$
3	$3.080\mathrm{e}{-01}\pm 1.510\mathrm{e}{-02}$	$1.328\mathrm{e}{-02}\pm 1.680\mathrm{e}{-03}$	$2.134\mathrm{e}{-01}\pm 2.560\mathrm{e}{-02}$
4	$2.959\mathrm{e}{-01}\pm 1.460\mathrm{e}{-02}$	$1.267\mathrm{e}{-02}\pm 3.110\mathrm{e}{-04}$	$1.994\mathrm{e}{-01}\pm 9.340\mathrm{e}{-03}$
5	$3.248\mathrm{e}{-01}\pm 9.690\mathrm{e}{-03}$	$1.358\mathrm{e}{-02}\pm 4.650\mathrm{e}{-04}$	$2.230\mathrm{e}{-01}\pm 1.100\mathrm{e}{-02}$

Appendix

Table 40: Effect of B-spline degree on PCA interpolation performance.

Degree	Wasserstein	MMD	Energy
1	$5.652\mathrm{e}{-01}\pm 2.770\mathrm{e}{-02}$	$3.352\mathrm{e}{-02}\pm 1.930\mathrm{e}{-03}$	$6.546\mathrm{e}{-01}\pm 5.310\mathrm{e}{-02}$
2	$5.861\mathrm{e}{-01}\pm 2.590\mathrm{e}{-02}$	$3.790\mathrm{e}{-02}\pm 2.500\mathrm{e}{-03}$	$7.244\mathrm{e}{-01}\pm 5.470\mathrm{e}{-02}$
3	$5.387\mathrm{e}{-01}\pm 3.350\mathrm{e}{-02}$	$3.583\mathrm{e}{-02}\pm 4.420\mathrm{e}{-03}$	$6.895\mathrm{e}{-01}\pm 1.010\mathrm{e}{-01}$
4	$5.389\mathrm{e}{-01}\pm 2.420\mathrm{e}{-02}$	$3.469\mathrm{e}{-02}\pm 2.410\mathrm{e}{-03}$	$6.588\mathrm{e}{-01}\pm 5.070\mathrm{e}{-02}$
5	$5.871\mathrm{e}{-01}\pm 1.240\mathrm{e}{-02}$	$3.842\mathrm{e}{-02}\pm 3.320\mathrm{e}{-03}$	$7.504\mathrm{e}{-01}\pm 5.620\mathrm{e}{-02}$

Appendix

Table 41: Effect of B-spline degree on PCA extrapolation performance.

$\bm{p}$	Model	Exp.	Damp.	LV	Lorenz
$0$	MMFM	$1.40\mathrm{e}{-3}$	$2.00\mathrm{e}{-3}$	$4.87\mathrm{e}{-1}$	$8.02\mathrm{e}{-1}$
$0$	SplineFlow ( $m=3$ )	$\mathbf{9.23\mathrm{e}{-5}}$	$\mathbf{3.10\mathrm{e}{-5}}$	$\mathbf{4.50\mathrm{e}{-2}}$	$\mathbf{6.39\mathrm{e}{-1}}$
$0.25$	MMFM	$2.00\mathrm{e}{-3}$	$1.80\mathrm{e}{-3}$	$3.72\mathrm{e}{-1}$	–
$0.25$	SplineFlow ( $m=3$ )	$\mathbf{8.42\mathrm{e}{-5}}$	$\mathbf{2.90\mathrm{e}{-5}}$	$\mathbf{1.87\mathrm{e}{-1}}$	–
$0.5$	MMFM	$1.70\mathrm{e}{-3}$	$3.00\mathrm{e}{-3}$	$4.41$	–
$0.5$	SplineFlow ( $m=3$ )	$\mathbf{1.13\mathrm{e}{-4}}$	$\mathbf{1.00\mathrm{e}{-4}}$	$\mathbf{2.75\mathrm{e}{-1}}$	–

Appendix

Table 42: Comparison of MMFM and SplineFlow across ODE systems and sampling irregularity

p

Flow Matching

Flow matching learns continuous-time vector fields that transport simple noise distributions into data for efficient generative modeling.

See all 656 papers

Oct 1, 2025

Multi-Marginal Flow Matching with Adversarially Learnt Interpolants

Learning the dynamics of a process given sampled observations at several time points is an important but difficult task in many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at discrete time steps, the problem of modelling the dynamics, and thus inferring the underlying trajectories, can be solved by multi-marginal generalisations of flow matching algorithms. This paper proposes a novel flow matching method that overcomes the limitations of existing multi-marginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss to fit neurally parametrised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. We showcase the versatility and scalability of our method by outperforming the existing baselines on spatial transcriptomics and cell tracking datasets, while performing on par with them on single-cell trajectory prediction. Code: https://github.com/mmacosha/adversarially-learned-interpolants.

Oskar Kviman, Kirill Tamogashev, Nicola Branchini +3

KTH · University of Edinburgh

Flow Matching Trajectory Inference

Ask PDF

Aug 23, 2026

I-SplineFlow: Learning Monotone Spline Stochastic Interpolant Schedulers for Few-Step Generation

Few-step generation with pretrained diffusion and flow models can be accelerated by lightweight training that optimizes the sampling trajectory rather than the network. A recent approach parameterizes the stochastic interpolant (SI) scheduler as a smooth curve whose control points enforce the three properties an SI scheduler must satisfy: fixed boundary conditions, a monotone signal-to-noise ratio (SNR), and differentiability. Existing parameterizations use globally supported polynomial bases, where every control point moves the whole curve and higher expressiveness needs a higher degree, which couples distant regions of the schedule during optimization. We introduce I-SplineFlow, which parameterizes the scheduler with integrated monotone splines (I-splines). I-splines decouple the polynomial degree from the number of mixture weights, so support width and smoothness can be chosen per model at a fixed weight count, and the compactly supported derivative basis makes the scheduler Jacobian orders of magnitude better conditioned than a Bézier basis. Boundary conditions and a strictly monotone SNR hold by construction, with no ordering constraint on the parameters and closed-form velocity derivatives. Across diffusion (EDM) and flow (ReFlow, Simple ReFlow) models, I-SplineFlow improves few-step FID over Bézier scheduling in most settings, most clearly at the lowest NFEs, and trains in minutes. Ablations show that both the degree freedom and the monotonicity constraint are needed. The code will be released upon acceptance.

Md Sakib Hossain Shovon, Md Rifat Ur Rahman, Md Abtahi Majeed Chowdhury +3

KAIST · Bangladesh University of Engineering and Technology

Few-Step Diffusion Sampling Diffusion Timestep Scheduling Stochastic Interpolants Flow-Based Generative Modeling

Ask PDF

Jun 2, 2026

Let the Dynamics Flow: Stable Flow Matching Dynamical Systems

Flow matching has recently emerged as a powerful approach for imitation learning, enabling scalable, expressive, and multimodal motion policies. However, when modeling these policies as dynamical systems, incorporating formal stability guarantees into these generative models is a prerequisite to ensure safe and generalizable robot behaviors, which remains a significant challenge. This paper introduces Stable Flow Matching Dynamical Systems (SFMDS), a novel framework that bridges the gap between highly expressive generative modeling and formal stability guarantees. SFMDS parametrizes dynamical systems via flow matching while constraining the model to satisfy positive invariance and/or Lyapunov stability conditions. We propose two variants: a soft constraint based on a penalty term, and a hard structural constraint embedded directly into the model architecture. We further extend both formulations to Lie groups to robustly handle orientation trajectories. Experiments on benchmark datasets, in simulation, and on a humanoid robot show that SFMDS learns stable, scalable, and multimodal dynamical systems in low- and high-dimensional state spaces, enabling safe and expressive robot motion generation. SFMDS matches or outperforms state-of-the-art methods on unimodal datasets, while substantially improving performance on multimodal datasets, where competing approaches fail to capture multi-modal behaviors. Accompanying source code and video are available at: https://let-the-dynamics-flow.github.io/SFMDS/.

Rodrigo Pérez-Dattari, Francisco Leiva, Andrea Testa +3

Department of Robotics, Perception, and Learning, KTH Royal Institute of Technology, Stockholm, Sweden. · Advanced Mining Technology Center (AMTC) and Department of Electrical Engineering, Universidad de Chile, Santiago, Chile. · Bosch Center for Artificial Intelligence, Renningen, Germany. +1

Flow Matching Lyapunov Stability Robot Motion Generation Robot Imitation Learning Flow-Based Generative Modeling

Ask PDF

Abstract

Flow Matching

Explore similar work

Multi-Marginal Flow Matching with Adversarially Learnt Interpolants

I-SplineFlow: Learning Monotone Spline Stochastic Interpolant Schedulers for Few-Step Generation

Let the Dynamics Flow: Stable Flow Matching Dynamical Systems