The researchers, led by Jingdi Wan and Yang Liu with colleagues including Hongping Wang, Bo Liu, Xiaolei Yang, Xiaodong Hu, Shengze Cai and Guowei He, report that existing machine-learning approaches fail to generalize the prediction of long-range particle trajectories when data are sparse. TrajectoryFlowNet is built to address this gap, combining Lagrangian and Eulerian perspectives on fluid motion within a physics-informed framework for fluid flow velocimetry and imaging.

According to the abstract, the model’s salient features include the ability to handle complex flow patterns with irregular boundaries, predict full-field flows, image the long-range flow trajectory of any arbitrary particle, and ensure physical consistency in predictions based on only scarce measurements of flow trajectories. This means the network can reconstruct detailed pictures of fluid motion even when only limited trajectory data are available, a regime where prior approaches struggled.

The team validated the architecture through both numerical and experimental test cases. On the numerical side, the examples included lid-driven cavity flow and complex cylinder flow, standard benchmarks that test a model’s capacity to capture diverse flow scenarios. On the experimental side, the study examined aortic and ventricle blood flows, extending the evaluation to real-world physiological conditions.

The reported results demonstrate the model’s effectiveness in capturing intricate particle-laden flow dynamics, enabling long-range tracking of particles and accurate construction of flow fields in real-world applications. The work is accompanied by five figures illustrating the overall TrajectoryFlowNet framework and the results of the cavity-flow, cylinder-flow, aortic blood-flow and left-ventricle blood-flow cases.

By learning to predict both the full flow field and individual particle trajectories from sparse measurements, the approach could reduce the experimental and computational burden traditionally associated with particle transport prediction in complex flows. The authors position the method as a step toward reliable flow imaging in settings where data are limited but physical consistency is essential.

Subject of Research: A Lagrangian–Eulerian physics-informed neural network for predicting flow fields and particle trajectories from sparse data

Article Title: Lagrangian–Eulerian learning of flow field and trajectories with TrajectoryFlowNet

Article References: Wan, J., Wang, H., Liu, B., Yang, X., Hu, X., Cai, S., He, G., & Liu, Y. (2026). Lagrangian–Eulerian learning of flow field and trajectories with TrajectoryFlowNet. Nature Machine Intelligence. https://doi.org/10.1038/s42256-026-01316-7

Image Credits: AI Generated

DOI: 10.1038/s42256-026-01316-7

Keywords: TrajectoryFlowNet, physics-informed neural network, fluid flow velocimetry, particle trajectories, Navier–Stokes equations, Lagrangian–Eulerian learning, blood flow imaging, lid-driven cavity flow, cylinder flow, sparse data, Nature Machine Intelligence, Lagrangian