TrajectoryFlowNet: Hybrid Flow & Particle Tracking

• TrajectoryFlowNet is a hybrid physics-informed model that couples Lagrangian particle tracking with Eulerian flow-field inference to solve inverse fluid dynamics problems.
• It enforces kinematic and dynamic consistency by integrating particle motion with Navier–Stokes constraints, enhancing prediction accuracy from sparse, noisy observations.
• The framework employs dual fully connected networks with Fourier feature mapping, validated on CFD benchmarks and biomedical flows to balance computational cost and data sparsity.

Searching arXiv for papers explicitly using or closely related to “TrajectoryFlowNet” to ground the article. TrajectoryFlowNet is a hybrid physics-informed neural framework for simultaneously reconstructing particle trajectories and inferring the underlying flow field from sparse passive-particle observations. In its explicit 2025 formulation, it addresses an inverse problem in fluid mechanics in which only a small number of tracer particles are observed, often at sparse time points and in noisy conditions, yet the objective is to recover both the full spatiotemporal velocity/pressure field and the motion of all particles. The method couples Lagrangian particle tracking with Eulerian flow-field inference through shared physical constraints, and is presented as a bridge between traditional numerical solvers and pure data-driven models (Wan et al., 13 Jul 2025).

1. Problem domain and conceptual scope

TrajectoryFlowNet is formulated for passive-particle tracking and flow-field inversion rather than for pedestrian forecasting or generic sequence modeling. The central setting is one in which sparse trajectories act as the primary observation modality, while the desired outputs are both the trajectories of all particles in the domain and the corresponding Eulerian flow state. The paper emphasizes that traditional numerical solvers require explicit governing equations, dense meshes, and large computational cost, whereas pure deep learning approaches can fit data without preserving physical consistency or generalization (Wan et al., 13 Jul 2025).

The model is designed around a hybrid Lagrangian–Eulerian view. In the Lagrangian description, a passive particle is treated as a fluid parcel identified by its initial position and release time; in the Eulerian description, the flow is represented at fixed spatial locations over time. TrajectoryFlowNet does not predict these two descriptions independently. Instead, it enforces a bidirectional coupling in which particle motion and flow-field reconstruction constrain one another.

This coupling is especially relevant in settings with irregular geometries, embedded obstacles, moving boundaries, and sparse measurements. The paper explicitly positions the method for particle tracking from sparse observations, flow-field inversion when direct velocity or pressure measurement is incomplete, biomedical imaging and cardiovascular flow analysis, and systems with sparse or noisy experimental data (Wan et al., 13 Jul 2025).

2. Hybrid Lagrangian–Eulerian formulation

The theoretical core of TrajectoryFlowNet is a coupled representation of particle motion and fluid dynamics. In the Lagrangian view, the trajectory is written as

$\mathbf{x}=\mathbf{x}(\boldsymbol{\xi},\tau),$

where $\tau=t-t_0$ is the particle’s motion time. The Eulerian view enters through the material derivative,

$$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$

so that

$$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$

The governing dynamics are constrained by the incompressible Navier–Stokes equation,

$$\frac{\partial u}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=-\frac{1}{\rho} \boldsymbol{\nabla} p+\frac{\mu}{\rho} \nabla^{2} \mathbf{u}.$$

The kinematic bridge between the two descriptions is the condition

$$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} \tau} = \mathbf{u},$$

which requires the particle velocity to equal the local flow velocity along the particle path (Wan et al., 13 Jul 2025).

This formulation makes the model an inverse solver with coupled latent structure rather than a trajectory regressor alone. Sparse particle observations anchor the Lagrangian description, while the Eulerian field is inferred over the full space-time domain through the learned mapping $(\mathbf{x},t)\mapsto(\mathbf{u},p)$. The paper argues that this is stronger than standard PINNs because it imposes PDE residuals while also integrating an explicit sequential trajectory-prediction mechanism (Wan et al., 13 Jul 2025).

3. Network architecture and spectral representation

TrajectoryFlowNet contains two fully connected subnetworks with distinct but linked roles. The first is the trajectory block $\mathcal{N}_1(\boldsymbol{\theta})$, which takes initial position $\mathbf{x}_0$, release time $t_0$, and motion time $\tau=t-t_0$0 as inputs, predicts a displacement, and forms the trajectory through

$\tau=t-t_0$1

Using automatic differentiation, it also yields Lagrangian velocity and acceleration:

$\tau=t-t_0$2

The second is the flow-field block $\tau=t-t_0$3, which takes trajectory points $\tau=t-t_0$4 and time $\tau=t-t_0$5 and outputs velocity and pressure,

$\tau=t-t_0$6

The two blocks are coupled by requiring the Eulerian velocity evaluated at the predicted trajectory to match the Lagrangian velocity:

$\tau=t-t_0$7

This equality is the central mechanism by which the trajectory network and the flow network constrain one another (Wan et al., 13 Jul 2025).

To mitigate spectral bias, TrajectoryFlowNet applies Fourier feature mapping to the inputs of both blocks:

$\tau=t-t_0$8

where $\tau=t-t_0$9 is a random Gaussian matrix with entries $$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$0. The stated purpose is to improve representation of complex, high-frequency, and unsteady flows, particularly those with sharp gradients or oscillatory structure (Wan et al., 13 Jul 2025).

4. Physics-informed constraints, loss construction, and optimization

The method is physics informed through two classes of constraints. The first is kinematic consistency, expressed by the requirement that particle motion and local fluid velocity coincide. The second is dynamic consistency, imposed through the incompressible Navier–Stokes residual:

$$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$1

The paper also gives the acceleration-level relation

$$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$2

The loss is divided into a data term and a physics term. The data loss is

$$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$3

The physics loss is

$$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$4

These terms are evaluated at data points and collocation points, and automatic differentiation is used to compute all derivatives at machine precision (Wan et al., 13 Jul 2025).

Training uses the L-BFGS optimizer with learning rate $$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$5 and up to 50,000 iterations on a machine with an Intel Xeon Platinum 8380 CPU and an NVIDIA A100 GPU. The reported model sizes vary by case; for example, the cavity-flow configuration uses a trajectory block with 4 layers $$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$6 40 neurons and a flow block with 6 layers $$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$7 60 neurons, while the cylinder and aortic cases use deeper configurations (Wan et al., 13 Jul 2025).

5. Validation cases and reported empirical behavior

The paper validates TrajectoryFlowNet on four scenarios spanning canonical CFD benchmarks and experimental cardiovascular flows. The reported results emphasize both trajectory fidelity and field-reconstruction quality (Wan et al., 13 Jul 2025).

Case	Data/setup	Reported outcomes
Lid-driven cavity flow	Side length 1; top wall moving at $$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$8 m/s; Reynolds number 100; 300 particles, with 200 for training and 100 for test	Trajectory error mean below 0.003; standard deviation under 0.0051; RMSE and MAE both below 0.011; correlations $$\frac{\partial}{\partial \tau}(f(\mathbf{x}(\boldsymbol{\xi}, \tau), t(\tau)))=\mathbf{u} \cdot \boldsymbol{\nabla} f+f_{t},$$9, $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$0, $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$1, $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$2, $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$3
Complex cylinder flow	2D duct with internal cylinder; inlet velocity 0.2 m/s; outlet pressure 0 Pa; Reynolds number 146,404; 2200 tracer particles	Trajectory error mean below 0.014; standard deviation under 0.0021; $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$4; $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$5; correlations $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$6, $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$7, $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$8, $$D_{\tau}=\frac{\partial}{\partial \tau}=(\partial_t+\mathbf{u}\cdot \boldsymbol{\nabla}).$$9, $$\frac{\partial u}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=-\frac{1}{\rho} \boldsymbol{\nabla} p+\frac{\mu}{\rho} \nabla^{2} \mathbf{u}.$$0
Experimental aortic blood flow	Real PIV measurements in a silicone aorta model following balloon-expandable TAVR; 50 time steps ($$\frac{\partial u}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=-\frac{1}{\rho} \boldsymbol{\nabla} p+\frac{\mu}{\rho} \nabla^{2} \mathbf{u}.$$1 s)	Correlations for passive particle trajectories and velocities exceeded 0.9; local prediction accuracy was reduced compared with synthetic cases
Experimental left ventricle blood flow	18,244 training points; fewer than 92 points per time step; 1,000 epochs; 200 time steps ($$\frac{\partial u}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=-\frac{1}{\rho} \boldsymbol{\nabla} p+\frac{\mu}{\rho} \nabla^{2} \mathbf{u}.$$2 s) predicted	Correlations for passive particle trajectories and velocities exceeded 0.9; strong long-horizon trajectory and field prediction despite extremely sparse data

These cases support several specific claims made in the paper. First, the model can recover canonical vortex structure in lid-driven cavity flow. Second, it can learn external boundary features in a cylinder flow with an internal obstacle and reconstruct pressure even though pressure is not directly measured by PIV. Third, it can operate under moving boundaries in the aortic and left-ventricle cases, where the network is said to infer boundary evolution indirectly from sparse trajectory data (Wan et al., 13 Jul 2025).

The paper also reports that TrajectoryFlowNet outperformed alternatives involving different activation functions, separate networks for trajectories and fields, and the proposed integrated architecture. Although the detailed numeric baseline tables are not reproduced in the supplied material, the stated conclusion is that joint hybrid modeling improved consistency relative to decoupled or less physics-aware alternatives (Wan et al., 13 Jul 2025).

6. Limitations, practical considerations, and common misconceptions

TrajectoryFlowNet is not presented as a generic replacement for numerical fluid solvers. The training process is nontrivial, the fully connected layers can become a computational bottleneck, and the method still depends on the quality and spatial coverage of passive-particle observations. The paper explicitly notes that stagnant-flow regions are harder to predict because the model depends on passive-particle coverage, and that highly complex flows such as very high Reynolds number or thermally coupled flows remain future work (Wan et al., 13 Jul 2025).

The method also does not yet provide uncertainty quantification. This is a substantive limitation relative to probabilistic flow or trajectory models, particularly in applications where ambiguity in the inferred field matters as much as point accuracy. In real experimental settings, local accuracy can degrade when latent physical variables are missing from the training data; this is highlighted in the aortic case, where hidden variables reduced local prediction quality compared with synthetic benchmarks (Wan et al., 13 Jul 2025).

Several recurring misconceptions are clarified by the formulation itself. One is that TrajectoryFlowNet is a pure data-driven model; it is not, because its objective explicitly includes kinematic and Navier–Stokes residual terms. Another is that the trajectory and field components are merely parallel heads; they are not, because the two subnetworks are coupled through the equality between Lagrangian particle velocity and Eulerian velocity along the path. A further misconception is that the framework requires explicit boundary-condition specification in the same manner as classical solvers. The paper states instead that it does not require such specification in the same way, and can infer boundary evolution indirectly from sparse trajectory data, although this should not be read as a claim that boundary effects become irrelevant (Wan et al., 13 Jul 2025).

7. Relation to other flow-based trajectory models

The supplied literature suggests that the label “TrajectoryFlowNet” is not used uniformly. In the explicit title usage, it denotes the hybrid Lagrangian–Eulerian fluid-mechanics model described above (Wan et al., 13 Jul 2025). In adjacent summaries, however, the same or closely related label is also attached to several distinct families of flow-based trajectory methods, which creates a naming ambiguity rather than a single unified research line.

One neighboring line is conditional flow matching for robotics. T-CFM models both forecasting and planning as conditional trajectory generation from Gaussian noise to data via a learned time-varying vector field, uses a 1D Convolutional Temporal U-Net with FiLM conditioning, and reports 35% higher predictive accuracy, 142% improved planning performance, and up to $$\frac{\partial u}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=-\frac{1}{\rho} \boldsymbol{\nabla} p+\frac{\mu}{\rho} \nabla^{2} \mathbf{u}.$$3 speed-up compared to diffusion-based models (Ye et al., 2024). Another is flow-guided crowd-motion prediction: FlowMNO frames pedestrian motion as a Markovian stochastic dynamical system, predicts the next optical-flow field from the current one, and integrates those predictions with Generalized Velocity Obstacles for robot navigation, with a reported average ADE reduction of $$\frac{\partial u}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=-\frac{1}{\rho} \boldsymbol{\nabla} p+\frac{\mu}{\rho} \nabla^{2} \mathbf{u}.$$4 across several baselines (Bhaskara et al., 2023).

A second cluster consists of normalizing-flow models for trajectory densities. FlowChain estimates future spatial densities analytically through a stack of conditional continuously-indexed flows, avoids KDE-based density estimation, and supports updates in less than one millisecond by reusing flow transformations and log-det-Jacobians (Maeda et al., 2023). TrajFlow appears in two distinct formulations: one learns distributions over latent abstractions of future trajectories using an RNN autoencoder plus conditional normalizing flow (Mészáros et al., 2023), while another models the marginal density of future spatial locations for occupancy estimation and implements both discrete and continuous variants using GRUs, neural CDEs, affine coupling layers, and neural ODE-based continuous normalizing flows (Kosieradzki et al., 24 Jan 2025).

A third group uses flow mechanisms for structured spatio-temporal prediction or policy learning. MotionFlow applies conditional normalizing flows with autoregressive masked-convolutional conditioning and time-factorized latent priors to structured motion prediction, including multi-agent trajectory forecasting (Zand et al., 2021). HBA-Flow introduces a Haar wavelet based block autoregressive flow for multimodal trajectory prediction with exact likelihood and $$\frac{\partial u}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=-\frac{1}{\rho} \boldsymbol{\nabla} p+\frac{\mu}{\rho} \nabla^{2} \mathbf{u}.$$5 sampling complexity (Bhattacharyya et al., 2020). Trajectory-Consistent Flow Matching, in turn, addresses the train–inference mismatch in flow-based visuomotor policies through auxiliary rectified flow velocity regression, multi-step trajectory consistency training, velocity-field regularization, and RK4 inference, achieving 70% and 60% overall success on two long-horizon multi-phase real-robot tasks where both baselines score 0% (Ahmed et al., 8 May 2026).

Within this broader landscape, the defining characteristic of the explicitly titled TrajectoryFlowNet is its direct coupling of passive-particle trajectories and Eulerian fluid fields under physics constraints. This distinguishes it from flow-matching planners, occupancy-density estimators, and generic probabilistic trajectory forecasters, even though all of these methods belong to the wider family of flow-based modeling approaches (Wan et al., 13 Jul 2025).