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

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 an 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 existing flow reconstruction methods, while the T2F+PINN model reduces the normalized error in vorticity reconstruction relative to the T2F model. We then apply the models in a turbulent Rayleigh-B\'{e}nard convection at a Rayleigh number of $Ra = 10<sup>{8}$ 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.