Lagrangian Attention Tensor Networks for Velocity Gradient Statistical Modeling

Direct numerical simulation of turbulence at realistic Reynolds numbers is still beyond current computational capability, necessitating models that reduce the number of resolved spatial scales. Motivated by phenomenology and recent data-driven works based on universality of the smallest scales in fully developed turbulence, the statistical dynamics of the velocity gradient tensor (VGT) at the Kolmogorov scale become of critical importance in advancing turbulence models. Physics-informed machine learning has found considerable success in exploiting large datasets taken from direct numerical simulation of Navier-Stokes to improve models for the evolution of the VGT. In this work, we follow the long line of blending physical insight with data analysis to simultaneously advance both the modeling and understanding of the phenomenology of the VGT. Using the intimate connection between VGT evolution and fluid deformation, we develop the Lagrangian attention tensor network approach that significantly improves over current physics-informed machine learning methods. We demonstrate state-of-the-art performance in both a-priori and a-posteriori metrics, before interpreting the trained attention mechanisms to discover a surprising connection between the history of the strain-rate-tensor and the pressure Hessian.