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Recurrent graph optimal transport for learning 3D flow motion in particle tracking


Flow visualization technologies such as particle tracking velocimetry are broadly used for studying three-dimensional turbulent flow in natural and industrial processes. Despite the advances in three-dimensional acquisition techniques, it is challenging to develop motion estimation algorithms in particle tracking due to large particle displacements, dense particle distributions and high computational cost. We present an end-to-end solution called graph optimal transport (GotFlow3D) to learn the three-dimensional fluid flow motion from consecutive particle images. The proposed model uses a graph neural network to extract geometric features and to further enrich the particle representations. The extracted deep features are subsequently used to correspond particles between consecutive frames, and the flow motion is then iteratively updated with a recurrent neural network approach. Experimental evaluations—including assessments on numerical experiments and validations on real-world experiments—demonstrate that GotFlow3D achieves state-of-the-art performance compared with recently developed scene flow learners and particle tracking algorithms. We believe that the high accuracy, robustness and generalization ability of our method can provide deeper insight into the complex dynamics of many physical and biological systems.

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Fig. 1: Overview of GotFlow3D for learning flow motion from particle sets.
Fig. 2: Flow estimation performance in FluidFlow3D-norm dataset.
Fig. 3: Robustness assessment for GotFlow3D.
Fig. 4: GotFlow3D applied to particle tracking velocimetry for CylinderFlow and DeformationFlow datasets.

Data availability

The synthetic dataset FluidFlow3D is available at The open-source dataset CylinderFlow is available at The open-source dataset DeformationFlow is available at

Code availability

The source code is available at


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This work was supported in part by: the National Key R&D Program of China (grant no. 2019YFB1705800); the National Natural Science Foundation of China (grant no. 61973270); and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (grant no. 61621002). S.C. gratefully acknowledges the support of Zhejiang University Education Foundation Qizhen Scholar Foundation.

Author information

Authors and Affiliations



J.L., C.X. and S.C. designed research, and edited and reviewed the paper. J.L. wrote the original draft and performed the research (implemented the algorithms and generated synthetic datasets). J.L. and S.C. analysed data. C.X. and S.C. supervised research.

Corresponding authors

Correspondence to Chao Xu or Shengze Cai.

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The authors declare no competing interests.

Peer review

Peer review information

Nature Machine Intelligence thanks Steven Wereley and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Robustness assessments for different flow learning methods with respect to (a) displacement ratio ρ, (b) noise from particles correspondence, (c) noise from particles localization.

The Acc Strict, Acc Relax, and Outliers are demonstrated as the evaluation metrics here. The proposed GotFlow3D outperforms the others in all the assessments.

Extended Data Table 1 Configuration of various flow cases in the synthetic dataset
Extended Data Table 2 Comparison between the state-of-the-art scene flow learning methods and GotFlow3D on different flow cases in the synthetic FluidFlow3D-norm dataset

Supplementary information

Supplementary Information

Ablation study on different modules in GotFlow3D, additional results on CylinderFlow dataset and GotFlow3D.

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Liang, J., Xu, C. & Cai, S. Recurrent graph optimal transport for learning 3D flow motion in particle tracking. Nat Mach Intell 5, 505–517 (2023).

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