End-to-end differentiable learning of turbulence models from indirect observations

The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields.The framework consists of a Reynolds-averaged Navier–Stokes(RANS)solver and a neuralnetwork-represented turbulence model,each accompanied by its derivative computations.For computing the sensitivities of the indirect observations to the Reynolds stress field,we use the continuous adjoint equations for the RANS equations,while the gradient of the neural network is obtained via its built-in automatic differentiation capability.We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures.We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists.The trained deep-neural-network turbulence model showed predictive capability on similar flows.

Frame Invariance and Scalability of Neural Operators for Partial Differential Equations

Partial differential equations(PDEs)play a dominant role in themathematicalmodeling ofmany complex dynamical processes.Solving these PDEs often requires prohibitively high computational costs,especially when multiple evaluations must be made for different parameters or conditions.After training,neural operators can provide PDEs solutions significantly faster than traditional PDE solvers.In this work,invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity.Neural operator based on graph kernel network(GKN)operates on graph-structured data to incorporate nonlocal dependencies.Here we propose a modified formulation of GKN to achieve frame invariance.Vector cloud neural network(VCNN)is an alternate neural operator with embedded frame invariance which operates on point cloud data.GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN.However,GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.

Seeing permeability from images： fast prediction with convolutional neural networks

Fast prediction of permeability directly from images enabled by image recognition neural networks is a novel pore-scale modeling method that has a great potential. This article presents a framework that includes (1) generation of porous media samples,(2) computation of permeability via fluid dynamics simulations,(3) training of convolutional neural networks (CNN) with simulated data, and (4) validations against simulations. Comparison of machine learning results and the ground truths suggests excellent predictive performance across a wide range of porosities and pore geometries, especially for those with dilated pores. Owning to such heterogeneity, the permeability cannot be estimated using the conventional Kozeny–Carman approach. Computational time was reduced by several orders of magnitude compared to fluid dynamic simulations. We found that, by including physical parameters that are known to affect permeability into the neural network, the physics-informed CNN generated better results than regular CNN. However, improvements vary with implemented heterogeneity.

Recent progress in augmenting turbulence models with physics-informed machine learning

In view of the long stagnation in traditional turbulence modeling,researchers have attempted using machine learning to augment turbulence models.This paper presents some of the recent progresses in our group on augmenting turbulence models with physics-informed machine learning.We also discuss our works on ensemble-based field inversion to provide training data for constructing machine learning models.Future and on-going research efforts are introduced.