Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents

High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g. long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy--Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples

Data-assisted reduced-order modeling of extreme events in complex dynamical systems

Dynamical systems with high intrinsic dimensionality are often characterized by extreme events having the form of rare transitions several standard deviations away from the mean. For such systems, order-reduction methods through projection of the governing equations have limited applicability due to the large intrinsic dimensionality of the underlying attractor but also the complexity of the transient events. An alternative approach is data-driven techniques that aim to quantify the dynamics of specific modes utilizing data-streams. Several of these approaches have improved performance by expanding the state representation using delayed coordinates. However, such strategies are limited in regions of the phase space where there is a small amount of data available, as is the case for extreme events. In this work, we develop a blended framework that integrates an imperfect model, obtained from projecting equations into a subspace that still contains crucial dynamical information, with data-streams through a recurrent neural network (RNN) architecture. In particular, we employ the long-short-term memory (LSTM), to model portions of the dynamics which cannot be accounted by the equations. The RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected in the reduced-order space. In this way, the data-driven model improves the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system dynamics. We assess the developed framework on two challenging prototype systems exhibiting extreme events and show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. The improvement is more significant in regions associated with extreme events, where data is sparse.Comment: Submitted to PLOS ONE on March 8, 201

A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities

We consider the problem of the probabilistic quantification of dynamical systems that have heavy-tailed characteristics. These heavy-tailed features are associated with rare transient responses due to the occurrence of internal instabilities. Systems with these properties can be found in a variety of areas including mechanics, fluids, and waves. Here we develop a computational method, a probabilistic decomposition-synthesis technique, that takes into account the nature of internal instabilities to inexpensively determine the non-Gaussian probability density function for any arbitrary quantity of interest. Our approach relies on the decomposition of the statistics into a 'non-extreme core', typically Gaussian, and a heavy-tailed component. This decomposition is in full correspondence with a partition of the phase space into a 'stable' region where we have no internal instabilities, and a region where non-linear instabilities lead to rare transitions with high probability. We quantify the statistics in the stable region using a Gaussian approximation approach, while the non-Gaussian distribution associated with the intermittently unstable regions of phase space is inexpensively computed through order-reduction methods that take into account the strongly nonlinear character of the dynamics. The probabilistic information in the two domains is analytically synthesized through a total probability argument. The proposed approach allows for the accurate quantification of non-Gaussian tails at more than 10 standard deviations, at a fraction of the cost associated with the direct Monte-Carlo simulations. We demonstrate the probabilistic decomposition-synthesis method for rare events for two dynamical systems exhibiting extreme events: a twodegree-of-freedom system of nonlinearly coupled oscillators, and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves

Data-driven sub-grid model development for large eddy simulations of turbulence

Turbulence modeling remains an active area of research due to its significant impact on a diverse set of challenges such as those pertaining to the aerospace and geophysical communities. Researchers continue to search for modeling strategies that improve the representation of high-wavenumber content in practical computational fluid dynamics applications. The recent successes of machine learning in the physical sciences have motivated a number of studies into the modeling of turbulence from a data-driven point of view. In this research, we utilize physics-informed machine learning to reconstruct the effect of unresolved frequencies (i.e., small-scale turbulence) on grid-resolved flow-variables obtained through large eddy simulation. In general, it is seen that the successful development of any data-driven strategy relies on two phases - learning and a-posteriori deployment. The former requires the synthesis of labeled data from direct numerical simulations of our target phenomenon whereas the latter requires the development of stability preserving modifications instead of a direct deployment of learning predictions. These stability preserving techniques may be through prediction modulation - where learning outputs are deployed via an intermediate statistical truncation. They may also be through the utilization of model classifiers where the traditional $L_2$-minimization strategy is avoided for a categorical cross-entropy error which flags for the most stable model deployment at a point on the computational grid. In this thesis, we outline several investigations utilizing the aforementioned philosophies and come to the conclusion that sub-grid turbulence models built through the utilization of machine learning are capable of recovering viable statistical trends in stabilized a-posteriori deployments for Kraichnan and Kolmogorov turbulence. Therefore, they represent a promising tool for the generation of closures that may be utilized in flows that belong to different configurations and have different sub-grid modeling requirements

Reduced flow models from a stochastic Navier-Stokes representation

International audienceIn large-scale Fluids Dynamics systems, the velocity lives in a broad range of scales. To be able to simulate its large-scale component, the flow can be decomposed into a finite variation process, which represents a smooth large-scale velocity component, and a martingale part, associated to the highly oscillating small-scale velocities. Within this general framework, a stochastic representation of the Navier-Stokes equations can be derived, based on physical conservation laws. In this equation, a diffusive sub-grid tensor appears naturally and generalizes classical sub-grid tensors. Here, a dimensionally reduced large-scale simulation is performed. A Galerkin projection of our Navier-Stokes equation is done on a Proper Orthogonal Decomposition basis. In our approach of the POD, the resolved temporal modes are differentiable with respect to time, whereas the unresolved temporal modes are assumed to be decorrelated in time. The corresponding reduced stochastic model enables to simulate, at low computational cost, the resolved temporal modes. It allows taking into account the possibly time-dependent, inhomoge-neous and anisotropic covariance of the small scale velocity. We proposed two ways of estimating such contributions in the context of POD-Galerkin. This method has proved successful to reconstruct energetic Chronos for a wake flow at Reynolds 3900, even with a large time step, whereas standard POD-Galerkin diverged systematically. This paper describes the principles of our stochastic Navier-Stokes equation, together with the estimation approaches, elaborated for the model reduction strategy