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

Yuhao Sun; Zekun Wu; Zixun Huang; Peijie Zhou

cs.LGAug 21, 2026

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

Authors: Yuhao Sun, Zekun Wu, Zixun Huang, Peijie Zhou

Organizations: Center for Machine Learning Research, Peking University · School of Mathematical Sciences, Peking University · Center for Quantitative Biology, Peking University · National Engineering Laboratory for Big Data Analysis and Applications, Beijing ·

Abstract

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.

Figures & tables

Figure 1: An illustration figure for TracingFlow

Method	Dynamics	2D Simulation		Cite 5D		Cite 100D
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	1st Order	0.5270 $\pm$ 0.0612	0.5922 $\pm$ 0.0813	0.8298 $\pm$ 0.0162	0.9155 $\pm$ 0.0160	10.5498 $\pm$ 0.0291	10.6220 $\pm$ 0.0310
SF 2 M	1st Order	0.9926 $\pm$ 0.1857	1.1318 $\pm$ 0.1734	0.7650 $\pm$ 0.0375	0.8956 $\pm$ 0.0408	12.3426 $\pm$ 0.0735	12.4757 $\pm$ 0.0783
3MSBM	2nd Order	1.1855 $\pm$ 0.2636	1.2767 $\pm$ 0.2215	3.6369 $\pm$ 0.3508	3.7703 $\pm$ 0.3178	15.7708 $\pm$ 0.1473	15.8833 $\pm$ 0.1832
MMFM		2.3689 $\pm$ 0.1830	2.6466 $\pm$ 0.1769	2.5346 $\pm$ 0.1520	2.7576 $\pm$ 0.1877	12.5353 $\pm$ 0.6268	12.7868 $\pm$ 0.7061
HRF		1.5390 $\pm$ 0.1130	1.6631 $\pm$ 0.1214	1.4016 $\pm$ 0.0437	1.5313 $\pm$ 0.0430	10.1532 $\pm$ 0.0643	10.2336 $\pm$ 0.0707

Table 1: Average

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between the generated and ground truth distributions at various time points on the 2D Simulation , Cite 5D, and Cite 100D datasets for different algorithms.

Figure 2: On the Simulation 2D dataset: a) non-crossing paths learned by OT-CFM, and b) crossing paths learned by TracingFlow.

Table 4

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Dynamics	$t=1$		$t=2$		$t=3$		$t=4$
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	1st Order	0.1215 $\pm$ 0.0164	0.1441 $\pm$ 0.0164	0.3644 $\pm$ 0.1177	0.4143 $\pm$ 0.1168	0.5863 $\pm$ 0.2615	0.6238 $\pm$ 0.2657	1.0356 $\pm$ 0.2325	1.1867 $\pm$ 0.2890
SF 2 M	1st Order	0.2389 $\pm$ 0.1217	0.2668 $\pm$ 0.1305	0.5700 $\pm$ 0.2574	0.6108 $\pm$ 0.2654	1.0630 $\pm$ 0.2769	1.3557 $\pm$ 0.2770	2.0986 $\pm$ 1.3198	2.2938 $\pm$ 1.3193
3MSBM	2nd Order	1.4029 $\pm$ 0.6892	1.4459 $\pm$ 0.6596	0.9318 $\pm$ 0.2553	1.0181 $\pm$ 0.2050	1.8079 $\pm$ 0.7761	1.9162 $\pm$ 0.7529	0.5993 $\pm$ 0.6176	0.7265 $\pm$ 0.6935
MMFM		1.3226 $\pm$ 0.1774	1.9287 $\pm$ 0.3002	1.8387 $\pm$ 0.5624	1.1779 $\pm$ 0.5114	1.8158 $\pm$ 0.1794	1.8740 $\pm$ 0.1701	4.4984 $\pm$ 0.2937	4.6059 $\pm$ 0.2977
HRF		1.2036 $\pm$ 0.0425	1.2096 $\pm$ 0.0244	0.9955 $\pm$ 0.0921	1.0009 $\pm$ 0.0873	1.9829 $\pm$ 0.1311	2.0040 $\pm$ 0.1582	1.9740 $\pm$ 0.1873	2.4379 $\pm$ 0.2157

Appendix

Table 4:

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between the generated and ground-truth distributions at each time point on the 2D Simulation dataset.

Method	Dynamics	$t=1$		$t=2$		$t=3$
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	1st Order	0.5615 $\pm$ 0.0156	0.6484 $\pm$ 0.0268	0.8472 $\pm$ 0.0463	0.9416 $\pm$ 0.0443	1.0536 $\pm$ 0.0534	1.1566 $\pm$ 0.0573
SF 2 M	1st Order	0.5932 $\pm$ 0.0249	0.6797 $\pm$ 0.0424	0.7768 $\pm$ 0.0451	0.8670 $\pm$ 0.0480	0.9251 $\pm$ 0.0279	1.0221 $\pm$ 0.0352
3MSBM	2nd Order	5.3735 $\pm$ 0.3961	5.4244 $\pm$ 0.3582	4.3118 $\pm$ 0.4589	4.5458 $\pm$ 0.4141	1.2254 $\pm$ 0.2998	1.3427 $\pm$ 0.3307
MMFM		0.5335 $\pm$ 0.0134	0.5911 $\pm$ 0.0057	2.2290 $\pm$ 0.2151	2.4557 $\pm$ 0.1613	4.8413 $\pm$ 0.2538	5.2261 $\pm$ 0.4061
HRF		0.4225 $\pm$ 0.0152	0.4622 $\pm$ 0.0185	0.6227 $\pm$ 0.0421	0.6814 $\pm$ 0.0392	0.7638 $\pm$ 0.0738	0.8394 $\pm$ 0.0713

Appendix

Table 5:

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between the generated and ground-truth distributions at each time point on the Cite 5D dataset

Method	Dynamics	$t=1$		$t=2$		$t=3$
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	1st Order	10.1136 $\pm$ 0.0104	10.1873 $\pm$ 0.0135	10.2504 $\pm$ 0.0209	10.3145 $\pm$ 0.0201	11.2682 $\pm$ 0.0938	11.3642 $\pm$ 0.0997
SF 2 M	1st Order	10.3598 $\pm$ 0.0051	10.4448 $\pm$ 0.0053	11.4647 $\pm$ 0.0384	11.5456 $\pm$ 0.0402	15.2032 $\pm$ 0.1813	15.4367 $\pm$ 0.1949
3MSBM	2nd Order	19.9677 $\pm$ 0.1373	20.1015 $\pm$ 0.1105	15.3218 $\pm$ 0.1232	15.4089 $\pm$ 0.1343	12.0229 $\pm$ 0.2895	12.1395 $\pm$ 0.3621
MMFM		9.8726 $\pm$ 0.0383	9.9305 $\pm$ 0.0361	12.1374 $\pm$ 0.2422	12.3076 $\pm$ 0.2495	15.5960 $\pm$ 1.4972	16.1222 $\pm$ 2.0195
HRF		9.9660 $\pm$ 0.0215	10.0580 $\pm$ 0.0341	9.9451 $\pm$ 0.0632	9.9986 $\pm$ 0.1526	10.5485 $\pm$ 0.1082	10.6442 $\pm$ 0.1254

Appendix

Table 6:

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between the generated and ground-truth distributions at each time point on the Cite 100D dataset

Figure 6: Trajectories learned by TracingFlow on the (a) Cite 5D and (b) Cite 100D datasets, projected to 2D using PCA.

Method	Dynamics	$t=1$		$t=2$		$t=3$		$t=4$
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	1st Order	3.4335 $\pm$ 0.1465	3.8462 $\pm$ 0.1449	4.8404 $\pm$ 0.1496	5.5792 $\pm$ 0.1358	5.0713 $\pm$ 0.2839	5.6448 $\pm$ 0.2861	7.1368 $\pm$ 0.3799	7.6938 $\pm$ 0.4574
SF 2 M	1st Order	3.4204 $\pm$ 0.0637	3.6713 $\pm$ 0.0589	3.5511 $\pm$ 0.0394	3.9441 $\pm$ 0.0418	4.3175 $\pm$ 0.2517	4.8738 $\pm$ 0.3247	6.9516 $\pm$ 0.0616	7.4727 $\pm$ 0.0795
3MSBM	2nd Order	6.1086 $\pm$ 1.1215	6.5434 $\pm$ 1.2227	7.3626 $\pm$ 0.8043	8.0596 $\pm$ 1.0252	7.5862 $\pm$ 1.1170	8.1734 $\pm$ 0.7900	7.7029 $\pm$ 0.4241	7.1022 $\pm$ 0.3971
MMFM		10.7803 $\pm$ 0.6718	11.8376 $\pm$ 0.7823	5.7932 $\pm$ 0.2162	6.2242 $\pm$ 0.1828	9.3326 $\pm$ 0.5617	9.9837 $\pm$ 0.4994	13.2046 $\pm$ 2.1528	14.3464 $\pm$ 2.2982
TF (Ours)		3.5611 $\pm$ 0.0362	3.8930 $\pm$ 0.0243	3.5002 $\pm$ 0.0983	3.8035 $\pm$ 0.1086	3.5643 $\pm$ 0.1140	3.7947 $\pm$ 0.4123	7.3639 $\pm$ 0.1316	7.6523 $\pm$ 0.1976

Appendix

Table 7:

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between predicted and ground-truth distributions at held-out time points on the EB 5D dataset.

Figure 7: Predicted distributions by TracingFlow at four held-out time points, projected to 2D using PCA.

Method	Dynamics	$t=1$		$t=2$
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	1st Order	2.6080 $\pm$ 0.0029	2.6869 $\pm$ 0.0030	1.8686 $\pm$ 0.0382	1.9456 $\pm$ 0.0403
SF 2 M	1st Order	2.8262 $\pm$ 0.0409	2.8373 $\pm$ 0.0462	0.2247 $\pm$ 0.0617	0.2533 $\pm$ 0.0568
3MSBM	2nd Order	4.0141 $\pm$ 0.8788	4.2262 $\pm$ 0.8596	0.7548 $\pm$ 0.0726	0.6751 $\pm$ 0.0573
MMFM		1.8585 $\pm$ 0.0350	2.1618 $\pm$ 0.0496	3.5695 $\pm$ 0.5692	4.1028 $\pm$ 0.6037
HRF		2.9150 $\pm$ 0.2595	2.9265 $\pm$ 0.3901	0.6717 $\pm$ 0.5491	0.6905 $\pm$ 0.4143

Appendix

Table 8: lineage-weighted

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between the generated and ground-truth distributions at each time point on the 3D Simulation Lineage dataset

Method	Dynamics	$t=1$		$t=2$
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	1st Order	13.2972 $\pm$ 0.0804	13.7134 $\pm$ 0.0831	16.6918 $\pm$ 0.1973	17.0807 $\pm$ 0.2025
SF 2 M	1st Order	14.1427 $\pm$ 0.0341	14.4199 $\pm$ 0.0303	23.2913 $\pm$ 0.2590	23.5314 $\pm$ 0.2640
3MSBM	2nd Order	17.6487 $\pm$ 0.5250	17.8710 $\pm$ 0.5204	19.0139 $\pm$ 1.4418	19.4144 $\pm$ 1.3622
MMFM		18.4040 $\pm$ 2.8129	19.8538 $\pm$ 2.8906	36.0412 $\pm$ 3.0429	40.2282 $\pm$ 3.3934
HRF		14.1007 $\pm$ 0.0804	14.3122 $\pm$ 0.0831	16.8135 $\pm$ 0.1973	17.1154 $\pm$ 0.2025

Appendix

Table 9: lineage-weighted

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between the generated and ground-truth distributions at each time point on the Hematopoiesis dataset

Method	Dynamics	$t=1$		$t=2$		Average
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	First-order	9.8780	10.6132	10.7621	11.4466	10.3201	11.0299
SF 2 M	First-order	10.2352	10.8017	13.2507	13.6879	11.7430	12.2428
3MSBM	Second-order	12.8214	13.2501	20.0262	20.6374	16.4238	16.9438
MMFM		10.4305	10.8695	22.3269	23.5322	16.3787	17.2008
HRF		10.6233	11.4510	10.9570	11.6201	10.7902	11.5356

Appendix

Table 10:

\mathcal{W}{1}

and

\mathcal{W}{2}

distances between the generated and ground-truth distributions at each time point on the Hematopoiesis dataset

Figure 8: Original data (a), and data points generated by TracingFlow on the Hematopoiesis dataset without biological priors (b) and with biological priors (c), projected to 2D using UMAP.

Method	Dynamics	$t=1$		$t=2$		$t=3$		$t=4$
Method	Dynamics	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
OT-CFM	First-order	0.0336	0.0375	0.0791	0.0887	0.1367	0.1415	0.1409	0.1474
SF 2 M	First-order	0.0768	0.0917	0.0802	0.0964	0.0819	0.0968	0.0809	0.0987
3MSBM	Second-order	0.4407	0.6305	0.5565	0.7617	0.5887	0.9113	0.5231	1.0714
MMFM		0.0346	0.0364	0.2961	0.3008	1.0538	1.0571	1.8755	1.8766
HRF		0.6205	0.6217	1.2004	1.2011	1.5334	1.5337	1.4986	1.5013

Appendix

Table 11:

\mathcal{W}_{1}

and

\mathcal{W}_{2}

distances between the generated and ground-truth distributions at each time point on the Mexico Gulf dataset

Method	Simulation	Cite 5D	Cite 100D	SimLineage-3D	RealLineage
OT-CFM	12 s	175 s	218 s	13 s	196 s
SF 2 M	26 s	79 s	84 s	34 s	61 s
MMFM	42 s	96 s	179 s	68 s	201 s
3 MSBM	220 s	495 s	910 s	244 s	855 s
CAF	94 s	274 s	382 s	256 s	264 s
HRF	247 s	271 s	359 s	289 s	347 s

Appendix

Table 12: Training time comparison (in seconds) of different methods across various datasets.

Batch Size	$t=1$		$t=2$		$t=3$		Average
Batch Size	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$	$\mathcal{W}_{1}$	$\mathcal{W}_{2}$
1000	0.4344	0.4798	0.6522	0.7183	0.7658	0.8479	0.6175	0.6820
2000	0.4196	0.4624	0.5640	0.6251	0.7972	0.8820	0.5936	0.6565
Full Dataset	0.2868	0.3185	0.5487	0.5997	0.7689	0.8499	0.5348	0.5894

Appendix

Table 13: Sensitivity analysis of batch size for Minibatch-OT on the Cite-5D dataset.

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Understanding cellular trajectories via time-resolved single-cell transcriptomics is vital for studying development, regeneration, and disease. A key challenge is inferring continuous trajectories from discrete snapshots. Biological complexity stems from stochastic cell fate decisions, temporal proliferation changes, and spatial environmental influences. Current methods often use deterministic interpolations treating cells in isolation, failing to capture the probabilistic branching, population shifts, and niche-dependent signaling driving real biological processes. We introduce Manifold Interpolating Optimal-Transport Flow (MIOFlow) 2.0. This framework learns biologically informed cellular trajectories by integrating manifold learning, optimal transport, and neural differential equations. It models three core processes: (1) stochasticity and branching via Neural Stochastic Differential Equations; (2) non-conservative population changes using a learned growth-rate model initialized with unbalanced optimal transport; and (3) environmental influence through a joint latent space unifying gene expression with spatial features like local cell type composition and signaling. By operating in a PHATE-distance matching autoencoder latent space, MIOFlow 2.0 ensures trajectories respect the data's intrinsic geometry. Empirical comparisons show expressive trajectory learning via neural differential equations outperforms existing generative models, including simulation-free flow matching. Validated on synthetic datasets, embryoid body differentiation, and spatially resolved axolotl brain regeneration, MIOFlow 2.0 improves trajectory accuracy and reveals hidden drivers of cellular transitions, like specific signaling niches. MIOFlow 2.0 thus bridges single-cell and spatial transcriptomics to uncover tissue-scale trajectories.

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