Generative stochastic modeling of strongly nonlinear flows with non-Gaussian statistics

Abstract

Strongly nonlinear systems, which commonly arise in turbulent flows and climate dynamics, are characterized by persistent and intermittent energy transfer between various spatial and temporal scales. These systems are difficult to model and analyze due to either high-dimensionality or uncertainty and there has been much recent interest in obtaining reduced models, for example in the form of stochastic closures, that can replicate their non-Gaussian statistics in many dimensions. On the other hand, data-driven methods, powered by machine learning and operator theoretic concepts, have shown great utility in modeling nonlinear dynamical systems with various degrees of complexity. Here we propose a data-driven framework to model stationary chaotic dynamical systems through non-linear transformations and a set of decoupled stochastic differential equations (SDEs). Specifically, we first use optimal transport to find a transformation from the distribution of time-series data to a multiplicative reference probability measure such as the standard normal distribution. Then we find the set of decoupled SDEs that admit the reference measure as the invariant measure, and also closely match the spectrum of the transformed data. As such, this framework represents the chaotic time series as the evolution of a stochastic system observed through the lens of a nonlinear map. We demonstrate the application of this framework in Lorenz-96 system, a 10-dimensional model of high-Reynolds cavity flow, and reanalysis climate data. These examples show that SDE models generated by this framework can reproduce the non-Gaussian statistics of systems with moderate dimensions (e.g. 10 and more), and approximate super-Gaussian tails that are not readily computable from the training data. Some advantageous features of our framework are convexity of the main optimization problem, flexibility in choice of reference measure and the SDE model, and interpretability in terms of interaction between variables and the underlying dynamical process.

Statistics of Extreme Events in Fluid Flows and Waves

“…Here we review a recent framework that focuses on the statistical extrapolation of the tails by using the data both as a source of information for dynamics and as a source of statistics about the system behavior (Arbabi & Sapsis 2019). The core idea is to use optimal transport of probabilities (Villani 2008, Parno & Marzouk 2018 to map the non-Gaussian data distribution to a reference measure, typically chosen to be Gaussian, and then model each dimension separately using a simple stochastic differential equation that mimics the spectrum of the time series (Figure 13).…”

“…The free parameters k j and β j are optimized for each system to match the power spectral density of v in order to make the model replicate the dynamics of the time series. We review results from Arbabi & Sapsis (2019) on climate data based on a six-hourly reanalysis of velocity and temperature in the Earth's atmosphere recorded at sigma level 0.95 from 1981 to 2017 (Berrisford et al 2011). These global fields were expanded in a spherical wavelet basis described by Lessig (2019) and the objective was to extrapolate the PDF tails for the u-velocity, U NP , and temperature, T NP , at the north pole.…”

Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning

“…Our setting is very different from identifying ODE from noisy observations, for example through Gaussian Processes [51]. Generative stochastic modeling of strongly nonlinear flows with non-Gaussian statistics is also possible [3]. Density and ensemble approximation techniques [50] provide a complementary view to the particle-like approach with SDE.…”

“…Normalize the matrix to mitigate effects of data density. 3. Compute eigenvectors to approximate eigenfunctions of ∆ evaluated on the data.…”

Nonlinear stochastic modeling with Langevin regression

“…However, NARMAX models typically cannot be transformed to continuous time, which is often the most natural setting for physical problems, and they often are black boxes that lack interpretability. More recent work has explored a variety of strategies for modeling nonlinear stochastic systems, including operator theoretic methods [19], optimal transport [20], deep learning [21,22,23], and identifying distribution evolution equations [24], although none of these pursues a representation in terms of nonlinear state-space dynamics. On the other hand, recent work has demonstrated that a stable linear system driven by colored noise can accurately reproduce the statistics of turbulence [25].…”