标题： 轨迹流网络：流动场和轨迹的混合拉格朗日-欧拉学习 显示英文标题

标题： TrajectoryFlowNet: Hybrid Lagrangian-Eulerian learning of flow field and trajectories

摘要： 流动携带粒子的过程非常复杂，传统上通过求解纳维-斯托克斯方程来处理。 尽管已经开发了不同的数值和实验技术，但这些方法需要对基础物理有深入的理解和\textcolor{black}{与高计算成本经常相关} 。 机器学习提供了一种新的替代方法，直接从数据中学习预测模式，从而绕过了显式物理建模的需要。 然而，纯粹的数据驱动方法有时可能缺乏可解释性和物理一致性。 通过将物理原理整合到机器学习中，可以弥合这一差距并解决上述问题。 在这一背景下，我们提出了TrajectoryFlowNet用于流动和粒子跟踪。 我们的方法结合了数据驱动学习的灵活性与基于物理的约束的严谨性，旨在实现准确性和效率。 我们模型的显著特点包括能够处理具有移动边界的复杂流动模式，预测域内所有粒子的轨迹，并仅基于稀疏轨迹确保预测过程中的物理一致性。 为了验证我们的方法，我们在一系列流动场景中进行了多个数值和实验案例。 这些实验展示了该模型在捕捉载粒流复杂动力学方面的有效性，推动了各种实际问题中精确的粒子跟踪和流动场反演。 显示英文摘要

摘要： The process of flows carrying particles is highly complex, traditionally tackled by solving the Navier-Stokes equations. Although different numerical and experimental techniques have been developed, these approaches demand a deep understanding of the underlying physics and \textcolor{black}{are frequently associated with high computational costs}. Machine learning offers a novel alternative, learning predictive patterns directly from data, thus bypassing the need for explicit physical modeling. Nonetheless, pure data-driven methods can sometimes lack interpretability and physical consistency. By integrating physics principles into machine learning, this gap can be bridged and the above problems can be solved. In this context, we have proposed TrajectoryFlowNet for flow and particle tracking. Our approach combines the flexibility of data-driven learning with the rigorousness of physics-based constraints, aiming to achieve both accuracy and efficiency. The salient features of our model include its ability to handle complex flow patterns with moving boundaries, predict the trajectories of all particles in the domain, and ensure physical consistency throughout the predictions based only on sparse trajectories. To validate our method, we have conducted several numerical and experimental cases across a range of flow scenarios. These experiments demonstrate the model's effectiveness in capturing the intricate dynamics of particle-laden flows, advancing precise particle tracking and flow field inversion in various real-world problems.