Session E31: Nonlinear Dynamics: Model Reduction & Turbulence I

Troubleshooting experiments using machine learning

Machine learning has already proven to be a useful tool for both data analysis and synthesis of mathematical models of complicated fluid systems directly from experimental measurements. This capability can be leveraged effectively to help troubleshoot experiments. While some types of experimental issues are easy to catch, others are too subtle to discover without an extensive and laborious analysis and often go undetected. In this talk we illustrate how slow variation in global parameters (e.g, associated with fluid leaks or drift in temperature or driving current) can be detected in an experimental setup generating a complicated (weakly turbulent) fluid flow in a thin layer of electrolyte driven by the Lorentz force. Machine learning is used to identify a set of governing equations (continuity and momentum balance), with a set of coefficients which depend on global parameters. Consequently, variation in the coefficients can be used to unravel the drift in the global parameters. This approach is widely applicable and can be used to diagnose a variety of other problems in various experimental setups by analyzing the data in real time or ex post facto.   

Physics-Guided Machine Learning Variational Multiscale Reduced Order Models

The variational multiscale framework (VMS) is motivated by the locality of energy transfer and enabled by the hierarchy of the underlying structures. Thus, VMS appears to be a natural solution for the closure problem in projection-based reduced order models (ROMs). Applications of VMS in ROM often focus on the use of phenomenological closure models by analogy with finite elements and large eddy simulations. Recently, data-driven VMS-ROMs have been explored, where a polynomial-like structure was considered to represent the interactions between different scales. We extend this by investigating whether neural networks can reveal the nonlinear processes and correlations as represented by the mutual interactions between various batches of resolved and unresolved scales. Moreover, we embed the locality of energy transfer into the learning and inference process of the neural network in a physics-guided machine learning (PGML) framework. We showcase the applicability of the proposed PGML-VMS-ROM using a set of prototypical flow problems with strong nonlinearity.   

Not Participating

In computational science and engineering, there is always a trade-off between the available computational resources and the desired level of accuracy. Therefore, in many complex multiphysics systems, solvers with varied levels of approximations are applied in different regions of the computational domain. One of the challenges with multi-fidelity computing is the accuracy of low-fidelity solvers and the recent advancement in machine learning can be utilized to build computationally cheap and accurate surrogate model. To this end, we present a coupled full order model (FOM) and reduced-order model (ROM) approach for considering the system of Boussinesq equations, which has application in various geophysical flows. In our approach, we employ a convolutional autoencoder network to recognize spatial patterns and eliminate nonlinear correlations among input features. Then, long short-term memory neural network architecture is utilized to discover temporal patterns in the low-rank space and complete the ROM part for representing the vorticity transport. The temperature evolution is treated in the FOM level, and we present a seamless coupling between FOM and ROM levels. The trade-off between accuracy and efficiency is analysed for solving a canonical Rayleigh-Benard convection system.   

On the connection between resolvent analysis and forced optimally time-dependent decomposition

This presentation addresses a new methodology for identifying the most responsive forcing and the most receptive states of a dynamical system. To this end, sensitivity analysis with forced optimally time-dependent (f-OTD) modes is introduced. With only the knowledge of the steady base flow or the mean flow, we establish a connection between f-OTD and resolvent analysis. The key observation is that the dominant f-OTD modes converge to the most responsive and forcing modes of the resolvent operator. We further demonstrate the ability of the f-OTD to compute the system response to an arbitrary forcing with an arbitrary time-dependent base flow. The theoretical results are demonstrated in two cases: the Burgers' equation and the temporally evolving jet.   

Nonlinear Reduced-Order Solutions for PDEs With Conserved Quantities: Applications to Fluid Dynamics

Reduced-order Nonlinear Solutions (RONS) is a new framework for reduced-order modeling of PDEs where the solution depends nonlinearly on time-varying variables [1]. RONS views all possible reduced solutions as a manifold embedded in the function space of the PDE. The time-dependent variables are evolved so that the instantaneous error between true dynamics of the PDE and dynamics of the reduced model are minimized. Additionally, in the RONS framework, any number of conserved quantities of the PDE can be easily enforced. In this talk, we demonstrate the application of RONS on several canonical problems: an advection-diffusion equation, the nonlinear Schrödinger equation for water waves, a modified nonlinear Schrödinger equation, and vortex dynamics in ideal fluids. In all scenarios, RONS accurately approximates the true solutions of the underlying PDEs at a low computational cost.   

No inverse necessary: a variational formulation of resolvent analysis

Resolvent analysis interprets the nonlinear term in the Navier-Stokes equations (NSE) as an intrinsic forcing to the linear dynamics. The conceptual picture of this phenomenon is inspired by control theory (CT), where the inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. However, the inversion of the linear operator inherent in the CT definition obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation. Additionally, for large systems this inversion leads to significant computational cost and memory requirements.  Here we suggest an alternative, inverse free, definition of the resolvent basis based on an extension of the Courant–Fischer–Weyl min-max principle in which resolvent response modes are defined as stationary points of a constrained variational problem.  This definition leads to a straightforward approach to approximate the resolvent modes of complex flows as expansions in any arbitrary basis. The proposed method avoids any matrix inversions and requires only the spectral decomposition of a matrix of significantly reduced size as compared to the original system.  This approach enables accurate reconstruction of the response modes regardless of the properties of the linear operator, however the non-normality and directional amplification of the NSE operator can lead to errors in the forcing modes and singular values. This variational framework is applied to a series of examples including a streamwise developing turbulent boundary layer to illustrate both the analytical and computational advantages of the proposed method.   

Modal analysis and interface tracking for multiphase flows

The primary breakup of a liquid core by an airblast atomizer is a complex phenomenon involving several instabilities that result in droplet generation and dispersion. We use back-lit imaging to distinguish the liquid-gas interface of a liquid-gas airblast atomizer at high temporal and spatial resolutions and employ Dynamic Mode Decomposition (DMD) to study the shape and frequency of instabilities of a liquid jet.  Because DMD is not suitable for interface tracking, we develop a data-driven two-step approach using the optical sensor data for the reconstruction and prediction of the location of the liquid-gas interface. The method uses DMD on the optical flow field estimated from image snapshot pairs. We demonstrate our method to a representative toy problem of an oscillating drop and on the primary atomization of a numerical planar liquid jet. Finally, we apply our method to the experimental liquid jet from the coaxial airblast atomizer using back-lit imaging. Our method is able to accurately reconstruct and predict the flow and preserves the sharpness of the fluid interface.   

A bi-fidelity framework to compute extreme-event probability

In this work, we propose a bi-fidelity sequential sampling framework to estimate the extreme-event probability. As a basis for sequential sampling, a bi-fidelity Gaussian process is used to infuse the high and low-fidelity samples to establish a surrogate model. A bi-fidelity acquisition function is proposed, which seeks a balance between the benefits and costs of adding high/low fidelity samples. This guides the selection of the next samples for both their location in parameter space and fidelity. We test this algorithm for both synthetic and real applications to demonstrate its effectiveness. For the latter, we consider a practical problem of estimating extreme ship motion probability in irregular waves using computational fluid dynamics (CFD) with two different grid resolutions.   

Machine Learning Statistical Evolution of the Coarse-Grained Velocity Gradient Tensor

We exploit recent advances in physics-informed machine learning and phenomenological theories of turbulence to develop parameterized stochastic differential equations (SDEs) coupling the Lagrangian evolution of a fluid volume to the coarse-grained velocity gradient tensor; resulting in a reduced order model for incompressible turbulence. Choosing minimal representations of fluid geometry and velocity gradient tensor, we search for local approximations to nonlinear (pressure and subgrid) terms. The goal is achieved by optimizing physics-informed neural networks - dependent on the coupled system of fluid geometry and velocity gradient tensor - over high fidelity Lagrangian direct numerical simulation data. We demonstrate the ability of the parameterized SDEs to reproduce the topological statistics of the coarse-grained velocity gradient tensor, as well as the shape distributions of the respective fluid elements.   

Dynamic-Mode Decomposition for Aero-Optic Wavefront Characterization

Aero-optical beam control relies on the development of low-latency forecasting techniques to quickly predict wavefronts aberrated by the turbulent boundary layer of an airborne optical system. We leverage the forecasting capabilities of the dynamic mode decomposition (DMD) – an equation-free, data-driven method for identifying coherent flow structures and their associated spatiotemporal dynamics – in order to estimate future state wavefront phase aberrations to feed into an adaptive optic control loop. We specifically leverage the optimized DMD (opt-DMD) algorithm on a subset of the Airborne Aero-Optical Laboratory Transonic (AAOL-T) experimental dataset. Critically, opt-DMD allows for de-biasing of the forecasting algorithm in order to produce a robust, stable, and accurate future-state prediction, and the underlying DMD algorithm provides a highly interpretable spatiotemporal decomposition of the turbulent boundary layer and the resulting aberrations to the wavefront dynamics.