TrajectoryFlowNet: Lagrangian-Eulerian learning of flow field and trajectories

TL;DR

TrajectoryFlowNet, a Lagrangian-Eulerian physics-informed neural network architecture, is proposed, a Lagrangian-Eulerian physics-informed neural network architecture, for fluid flow velocimetry and imaging via learning to predict spatiotemporal flow fields and long-range particle trajectories.

Abstract

Predicting particle transport in complex flows is traditionally achieved by solving the Navier-Stokes equations. While various numerical and experimental methods exist, they typically require deep physical insights and incur high computational costs. Machine learning offers an alternative by learning predictive patterns directly from data, avoiding explicit physical modeling. However, purely data-driven approaches often lack interpretability, physical consistency, and generalizability in sparse data regimes. To this end, we propose TrajectoryFlowNet, a Lagrangian-Eulerian physics-informed neural network architecture, for fluid flow velocimetry and imaging via learning to predict spatiotemporal flow fields and long-range particle trajectories. The salient features of our model include its ability to handle complex flow patterns with irregular boundaries, predict the full-field flows, image the long-range flow trajectory of any arbitrary particle, and ensure physical consistency in predictions based only on very scarce measurement of flow trajectories. We validate TrajectoryFlowNet via both numerical examples (e.g., lid-driven cavity flow and complex cylinder flow) and experimental test cases (e.g., aortic and ventricle blood flows) across diverse flow scenarios. The results demonstrate our 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.

Figures (9)

• Figure 1: Overview of TrajectoryFlowNet. a, Schematic of simultaneous prediction of particle trajectories and flow field. The computational domain contains only the initial positions of passive particles (red dots). The objective is to predict the trajectories of these particles (blue dots) and infer the spatiotemporal flow field. b, Schematic architecture of TrajectoryFlowNet, which consists of a Trajectory Block and a Flow Field Block. c, Schematic of the Trajectory Block (with trainable parameters $\boldsymbol{\theta}$), which maps the initial position, release time and motion time $\{\mathbf{x}_0,t_0,\tau\}$ of a particle to its displacement $\tilde{\mathbf{x}} = \tilde{\mathbf{x}}(\mathbf{x}_0, t_0, \tau)$ after Fourier feature mapping. By adding this displacement to the initial position, we obtain the particle's trajectory $\mathbf{x} = \tilde{\mathbf{x}} + \mathbf{x}_0$. d, Schematic of the Flow Field Block, controlled by trainable parameters $\boldsymbol{\vartheta}$, which takes the Fourier feature-mapped particle trajectories and motion times (predicted by Trajectory Block) as input and learns the mapping to the flow velocity $\mathbf{u}$ and pressure $p$. Using automatic differentiation (AD), we compute the derivative of the particle position $\mathbf{x}$ with respect to the particle motion time $\tau$ (aka, $\mathrm{d}\mathbf{x}/\mathrm{d} \tau$) at machine precision. The residual between this derivative and the network-predicted velocity is used to enhance physical consistency of the network. Note that the input-output variables are nondimensionlized. The predicted flow velocity and pressure fields are assumed to satisfy the Navier-Stokes (NS) equations, leading to the NS loss. The total loss function includes data losses $\mathcal{L}_x$ (trajectory loss) and $\mathcal{L}_v$ (velocity loss), as well as the residual physics losses $\mathcal{L}_{com}$ (compatibility loss) and $\mathcal{L}_{NS}$ (NS equation loss). Here, $\mathcal{D}_d$ represents the training data, and $\mathcal{D}_c$ the collocation points. $\lambda_1$, $\lambda_2$, $\lambda_3$ denote the loss hyperparameters.
• Figure 1: Comparison of TrajectoryFlowNet and AIV for flow field prediction ($u$, $v$, $p$) in the lid-driven cavity flow case at $t = 25$ s.
• Figure 2: Results of the Lid-driven Cavity Flow Case.a, Description of lid-driven cavity flow test setup. A $1\times1$ square cavity model was established, with fixed left, right, and bottom walls, and a moving top lid from left to right at an initial speed of 1 m/s, resulting in a flow with a Reynolds number of 100. The problem was simulated by FLUENT. At $t_0 = 5$ s, 300 particles are released within the cavity, and their trajectories, velocities, and pressure data are recorded. b, Illustration of typical particle trajectories in the training set. Red dots denote the initial particle positions, and gray dots the trajectory points. c, Illustration of randomly selected predicted particle trajectories in the test set. The red dots represent the initial positions of the passive particles, the orange solid lines the predicted trajectories by TrajectoryFlowNet, and the dashed lines the true values. d, The flow velocity magnitude field $t = 25$ s (from left to right: the velocity magnitudes of all passive particles in the training set, snapshot of the ground truth, snapshot of the prediction by TrajectoryFlowNet, and the prediction error). e, Temporal evolution of the relative $L^2$ error for the predicted particle trajectories in the test set. The black line represents the relative $L^2$ error in the $x$-direction, and the red line for the $y$-direction. Over the considered time period, the relative $L^2$ error remains below 1.3% in both directions, demonstrating the high accuracy and temporal consistency of the model’s predictions. f, Distribution of the predicted velocity magnitudes for the text set at $t = 25$ s via Kernel density estimation (KDE). The solid blue line represents the ground truth, and the dashed red line the prediction by TrajectoryFlowNet. e, KDE distribution of the predicted velocity magnitude errors at $t = 25$ s for the test set. The errors are normalized by the dynamic range of the respective velocity components as the characteristic scale.
• Figure 2: Comparison of TrajectoryFlowNet and AIV for flow field prediction ($u$, $v$, $p$) in the complex cylinder case at $t = 25$ s.
• Figure 3: Results of the Complex Cylinder Flow Case.a, Description of complex cylinder flow setup. The cylinder is placed inside a duct featuring an inlet and an outlet, with liquid water as the working fluid. A steady inlet velocity of $0.2$ m/s was employed, resulting in a Reynolds number of 146,404, with zero outlet pressure. At $t=10$ s, 2200 tracer particles are released at the inlet following by tracking their positions and velocities. This case was simulated by FLUENT. b, An example set of 300 particle trajectories in training set. Red dots denote initial particle positions, and gray dots the trajectory points. c, Snapshot of the flow velocity magnitudes of all particles of training set at $t = 25$ s. d, Illustration of randomly selected predicted particle trajectories in test set. Red dots marking the initial positions, orange solid lines the predicted trajectories, and dashed lines the true values. e, The flow velocity magnitude field at $t = 25$ s (from left to right: snapshot of the ground truth, snapshot of the prediction, and the corresponding prediction error). f, Temporal evolution of trajectory prediction errors in test set. The black line represents the relative $L^2$ error in the $x$-direction, the red line for the $y$-direction. g, KDE distribution of predicted velocity magnitudes in test set at $t = 25$ s. The solid blue line represents the ground truth, and the dashed red line the prediction by TrajectoryFlowNet. h, KDE distribution of normalized absolute errors for velocity magnitudes at $t = 25$ s in the test set.