Session A11: Turbulence: Modeling & Simulations I: Data-Driven and Machine Learning Approaches

Capturing small-scale dynamics of turbulence using deep learning

Turbulent flows are characterized by a wide range of interacting scales. While the large-scales are flow-dependent, the small-scales features such as the statistics derived from the velocity gradient tensor are known to display various universal properties. Hence, understanding and predicting the velocity gradient dynamics is of paramount importance for both theoretical progress and successful modeling. For modeling purposes, the pressure Hessian and viscous Laplacian result in unclosed terms, and have been the subject of various analytical closure approaches. In this work, we instead use a deep learning framework to model these terms by utilizing the tensor-based neural network (TBNN), which satisfies various symmetries and physical constraints by design. The TBNN is trained using a massive database generated using direct numerical simulations (DNS) of isotropic turbulence in periodic domains, of up to 12288³ grid points, with the Taylor-scale Reynolds number R_λ ranging from 140-1300. The resulting model shows good agreement with the DNS data for statistics which are not very sensitive to R_λ. By analyzing the statistics of pressure and viscous terms, we discuss strategies to explicitly incorporate R_λ-dependence in the model to capture the effects of intermittency.   

Data Driven Learning of Mori-Zwanzig Operators for Isotropic Turbulence

The Mori-Zwanzig (MZ) framework provides a mathematically formal procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Turbulence models based on MZ formalism have been scarce due to the limited knowledge of the MZ operators. In this work, we apply a recently developed data-driven learning algorithm on a set of fully-resolved isotropic turbulence datasets to extract the MZ operators. With data augmentation using known turbulence symmetries, the extracted Markov term, memory kernel, and orthogonal dynamics are statistically converged and the Generalized Fluctuation-Dissipation relation can be verified. The properties of the memory kernel and orthogonal dynamics, and their dependence on the choices of observables are investigated to shed light on turbulence physics and address the modeling assumptions that are commonly used in MZ-based models. A series of numerical experiments are then constructed to evaluate the memory effects on predictions. Results show that the prediction errors are strongly affected by the choice of observables and can be further reduced by including the past history of the observables in the memory kernel.   

Reconstruction of turbulent data from TURB-Rot database with deep generative models and Gappy POD

We study the applicability of tools developed by the computer vision community for feature learning and semantic image inpainting to perform data reconstruction of fluid turbulence configurations. The aim is twofold. First, we explore on a quantitative basis, the capability of Convolutional Neural Networks embedded in a Deep Generative Adversarial Model (Deep-GAN) to generate missing data in turbulence, a paradigmatic high dimensional chaotic system. In particular, we investigate their use in reconstructing two-dimensional damaged snapshots extracted from a large database of numerical configurations of 3d turbulence in the presence of rotation and of turbulent channel flows. The generative model we present is based on a first Context Encoders network that infers the missing data via minimization of the $L_2$ pixel-wise reconstruction loss, plus an adversarial penalization given by a second network that aims to discriminate real from reconstructed data. Finally, we present a comparison with different and well-known data assimilation tools, such as Nudging,  an equation-informed unbiased protocol, or on Gappy POD, developed in the context of reconstruction of images. The TURB-Rot database, http://smart-turb.roma2.infn.it, of 300K 2d turbulent images is describeM. Buzzicotti$^1$, T. Li$^2$, F. Bonaccorso$^{1,3}$,  P. Clark Di Leoni$^4$, L. Biferale$^1$d and details on how to download it are given.    

Non-intrusive sensing from coarse measurements by means of generative adversarial networks (GANs)

In this study we demonstrate the applicability of super-resolution generative adversarial networks (SRGANs) to reconstruct turbulent-flow quantities from coarse wall measurements. The method is applied both for resolution enhancement of wall data and for the estimation of wall-parallel velocity fields from coarse wall-shear stress and wall-pressure measurements. We illustrate the use of the method in a turbulent open-channel flow at a friction Reynolds number of $Re_{\tau} = 180$. We use a direct-numerical-simulation (DNS) database for training, and consider spatial downsampling factors equal to 4, 8 and 16 in each wall-parallel direction. Then we reconstruct wall-parallel fluctuation fields at inner-scaled wall-normal locations $y^+$ ranging from 15 to 100. We first show that SRGAN can be successfully used to enhance the resolution of the coarse wall measurements. Furthermore, this method can be used to perform the two steps combined ({\it i.e.}, super-resolution of wall information and flow prediction), obtaining very good reconstruction results. It is shown that even for the most challenging cases the SRGAN is capable of capturing the large-scale structures of the flow. This novel methodology is also applied to perform non-intrusive sensing in turbulent urban-flow environments.   

Data-driven subgrid-scale parameterization of turbulence in the small-data limit

In this work, we develop a data-driven subgrid-scale(SGS) model for large eddy simulation of turbulence using a fully convolutional neural network(CNN). We first conduct direct numerical simulation (DNS) and obtain training, validation, and testing data sets by applying a Gaussian spatial filter to the DNS solution. We train the CNN with the filtered state variables, i.e., vorticity and stream function as inputs and the nonlinear SGS term as an output. A priori analysis shows that the CNN-predicted SGS term accurately captures the inter-scale energy transfer. A posteriori analysis indicates that the LES-CNN outperforms the physics-based models in both short-term prediction and long-term statistics. In the small-data limit, the LES-CNN generates artificial instabilities and thus leads to unphysical results. We propose three remedies for the CNN to work in the small-data limit, i.e., data augmentation and group convolution neural network, leveraging the rotational equivariance of the SGS termand incorporating a physical constraint on the SGS enstrophy transfer. The SGS term is both translational and rotational equivariant in a square periodic flow field. While primitive CNN can capture the translational equivariance, the rotational equivariance can be accounted for by either including rotated snapshots in the training data set or by a GCNN that enforces rotational equivarianceas a hard constraint. Additionally, The SGS enstrophy transfer constraint can be implemented in the loss function of the CNN. A priori and a posteriori analyses show that the CNN/GCNN with knowledge/constraints of rotational equivariance and SGS enstrophy transfer enhances the SGS model and allows the data-driven model to work stably and accurately in a small-data limit.   

Wall-models of turbulent flows via scientific multi-agent reinforcement learning

We introduce a methodology for the semi-automated discovery of wall models for large-eddy simulations (LES). The methodology, scientific multi-agent reinforcement learning (SciMARL), fuses the numerical discertization of the flow governing equations with multi-agent reinforcement learning.  In SciMARL, the discretization points  act simultaneously  as cooperating agents that learn to supply the LES closure model. A particular advantage of  SciMARL over other machine learning methodologies is its generalisation capabilities with limited data. The agents self-learn closures as action policies that generalize to extreme Reynolds numbers and previously unseen geometries. The present simulations reduce by several orders of magnitude the computational cost over fully-resolved simulations while reproducing key flow quantities. We believe that SciMARL opens new capabilities for the modeling and simulation of turbulent flows.   

Machine Learning-based Model to Improve Wall-modeled Large-eddy Simulation of Supersonic Turbulent Flows

A machine learning algorithm is presented, serving as a data-driven modeling tool for wall-modeled large-eddy simulations (WMLES). The proposed model is formulated to address the problems of log layer mismatch (LLM) and inaccurate prediction of skin friction, particularly for supersonic separated and reattached flows. This machine learning algorithm uses Random Forest Regression (RFR) to map the local mean flow fields to the discrepancies in the skin friction (heat flux) while complying with Galilean invariance, as the flow features input is provided using relative velocities. The model is tested on two different supersonic flows, namely, flow over a flat plate and flow around an expansion-compression corner. The performance is evaluated by comparing the skin friction (heat flux) and flow properties with exact values. The ultimate goal is to build a robust and generalizable machine learning model to improve the prediction of WMLES of supersonic flows. To this end, the model is trained by a set of flows containing some essential flow physics to devise a generalizable model.   

Machine Learning Lagrangian Large Eddy Simulations with Smoothed Particle Hydrodynamics

In this work, we apply Physics-Informed Machine Learning to develop Lagrangian Large Eddy Simulation (LES) models for turbulent flows. We extend the weakly compressible Smoothed Particle Hydrodynamics (SPH) formalism using a broader set of parameterizations, by combining physics-based parameters and physics-inspired Neural Networks (NN) to describe the evolution of turbulence within the resolved range of scales. The sub-grid scale contributions, similar to those in LES, are modeled separately using NN with physical constraints to account for the effects from un-resolved scales.  We construct the resulting model under the Differentiable Programming framework to facilitate efficient training and then train the model on a set of coarse-grained Lagrangian data extracted from fully-resolved Direct Numerical Simulations. We experiment with loss functions of different types, including physics-informed ones accounting for statistics of Lagrangian particles. We show, through a series of diagnostic tests, that the developed model is capable of reproducing flow structures at the resolved scale and important Lagrangian and Eulerian statistics of turbulent flows.   

Reinforcement learning for autonomous navigation of swimmers in turbulent flow

Efficient navigation of autonomous swimmers is crucial for numerous applications, ranging from synthetic microswimmers for targeted drug delivery to oceanographic buoys for ocean/ weather monitoring and spilled oil tracking. In this paper, we study autonomous navigation within a turbulent flow, which is challenging due to the nonlinearity of the flow. In particular, a deep reinforcement learning (RL) technique is employed to train an autonomous swimmer to navigate efficiently towards a target in a two-dimensional turbulent flow by changing their flow direction. A neural network, that maps the measured state to possible actions, is trained by repeated experience with the turbulent flow environment. The resulting controller is compared to a 'naive' swimmer which is always directly oriented towards the target regardless of the underlying flow. The RL swimmer performs on average significantly better than the naive swimmer and learns to utilize vortical motion to its advantage by aligning its swimming direction with the flow direction.