Super-resolution reconstruction of turbulent flows from a single Lagrangian trajectory

TL;DR

This work addresses reconstructing high-resolution Eulerian flow fields from sparse, single Lagrangian trajectories by introducing track-to-flow (T2F), an encoder–decoder model with a Vision Transformer encoder and CNN decoder. It also presents a physics-informed variant, T2F+PINN, that augments data loss with PDE residuals to enforce dynamical consistency, improving gradient-related quantities such as vorticity and temperature gradients. Across a laminar cylinder wake (Re = 800) and turbulent Rayleigh–Bénard convection (Ra = 10^8, Pr = 0.71), T2F accurately recovers primitive fields while T2F+PINN yields substantial gains in gradient fidelity (up to ~60% improvements for Q and similar gains for ω_z and ∂xT), albeit with some trade-offs in primitive-variable accuracy. The results highlight the value of physics-informed constraints for gradient-rich reconstructions from sparse Lagrangian data and indicate the need for retraining to transfer across RB configurations, with potential for online/adaptive extensions in real-time sensing contexts.

Abstract

We studied the reconstruction of turbulent flow fields from trajectory data recorded by actively migrating Lagrangian agents. We propose a deep-learning model, track-to-flow (T2F), which employs a vision transformer as the encoder to capture the spatiotemporal features of a single agent trajectory, and a convolutional neural network as the decoder to reconstruct the flow field. To enhance the physical consistency of the T2F model, we further incorporate a physics-informed loss function inspired by the framework of physics-informed neural network (PINN), yielding a variant model referred to as T2F+PINN. We first evaluate both models in a laminar cylinder wake flow at a Reynolds number of $Re = 800$ as a proof of concept. The results show that the T2F model achieves velocity reconstruction accuracy comparable to that of existing flow reconstruction methods, while the T2F+PINN model reduces the normalised error in vorticity reconstruction relative to the T2F model. We then apply the models in a turbulent Rayleigh-Bénard convection at a Rayleigh number of $Ra = 10^8$ and a Prandtl number of $Pr = 0.71$. The results show that the T2F model accurately reconstructs both the velocity and temperature fields, whereas the T2F+PINN model further improves the reconstruction accuracy of gradient-related physical quantities, such as temperature gradients, vorticity and the Q value, with a maximum improvement of approximately 60 % compared to the T2F model. Overall, the T2F model is better suited for reconstructing primitive flow variables, while the T2F+PINN model provides advantages in reconstructing gradient-related quantities. Our models open a promising avenue for accurate flow reconstruction from a single Lagrangian trajectory.

Figures (21)

• Figure 1: Overview of the T2F model for reconstructing flow fields in the cylinder wake. An actively navigating Lagrangian agent collects local flow cues along its trajectory, which are subsequently used to infer the surrounding Eulerian flow field.
• Figure 2: Schematic of the T2F model architecture. The model consists of a ViT encoder that extracts spatiotemporal features from Lagrangian trajectory data, followed by a CNN decoder that reconstructs the corresponding Eulerian flow field.
• Figure 3: Trajectories of self-propelling agents navigating from the initial region (red) to the terminal region (blue) within the cylinder wake flow. The background contours represent the instantaneous out-of-plane vorticity field, illustrating the underlying flow structures guiding agent migration.
• Figure 4: Reconstruction results from the T2F and T2F+PINN models for a representative input in the cylinder wake. Ground-truth fields of (a) horizontal velocity $u_x^{\ast}$, (b) vertical velocity $u_y^{\ast}$ and (c) out-of-plane vorticity $\omega_z^{\ast}$. (d--f) Reconstructions by the T2F model. (g--i) Reconstructions by the T2F+PINN model. Listed values denote the normalised $L_2$ error $\epsilon$.
• Figure 5: Pointwise reconstruction errors of the horizontal velocity $u_x^{\ast}$, vertical velocity $u_y^{\ast}$ and vorticity $\omega_z^{\ast}$ in the cylinder wake: (a--c) the T2F model, (d--f) the T2F+PINN model.