Effective Data-Driven Models for Chaotic and Turbulent Dynamics

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Recent advances in computing algorithms and hardware have rekindled interest in developing high accuracy, low-cost reduced models for simulating complex physical systems. Such data-driven models allow us to overcome several difficult practical issues, such as (a) extreme computational requirements of direct numerical simulations of complex partial differential equations exhibiting multiple temporal and spatial scales; (b) insufficient data or unknown parameters characterizing sub-processes; and (c) no knowledge of the governing subsystem of equations. In the case of (a)(b), the form of the dynamics is often at least partially known and it is possible to parametrically estimate a reasonable expression that describes the time evolution of the dynamical process. In the first part of this thesis, we present the simulated least absolute shrinkage and selection operator (SLASSO), a statistical regression method, to select essential parameters from an assumed efficient stochastic model and obtain corresponding parameter estimations. Our developed approach is particularly suitable for data generated by large-scale variables in multiscale systems. The SLASSO estimator overcomes the issue of a large discretization error for data sets sub-sampled with a large observational time step. In contrast with traditional Maximum Likelihood Estimators and LASSO estimators for stochastic differential equations, the SLASSO approach is able to correctly select the model that fits the data sub-sampled with a large observational time step. We illustrate this approach on the multiscale additive triad model and the discretized Burgers model. Statistical properties of the slow/large scale variable are reproduced well for the additive triad model. However, the SLASSO estimator only produces adequate results when one has good knowledge of the mathematical expression of the dynamics. In addition, similar to many other parametric approaches, the SLASSO estimator is prone to overfitting. In the case of (a)(b)(c), the goal is to find a reduced model that replaces expensive direct numerical simulations of complex partial differential equations at fine spatial and temporal scales. In addition, it is often desirable to use observational data to infer a reduced model. To this end, we construct a reduced model using machine-learned surrogates that efficiently and accurately forecast trajectories and ensemble properties of the underlying system. In particular, we develop Reservoir Computing for accurate prediction of dynamical systems. Such models need to be trained on a reasonably large dataset generated by ``true'' dynamics, but later can be used for a fast trajectory forecast in an initial-value problem. We apply our approach to the one-dimensional shallow water equations, which is a classical model in fluid dynamics. We assume that we only know a finite set of samples of this system without knowledge of the mathematical structure of the shallow water systems. We introduce a data-driven approach: Echo State Network (ESN). This network can be trained by a fast, stable, and simple training algorithm via regularized linear regression. The ESN is able to efficiently and accurately forecast future system states of the shallow water equations (SWE). Moreover, we demonstrate that the ESN outperforms polynomial regression and is robust with respect to perturbations of the initial data. We illustrate the performance of our algorithms through extensive experiments. We also introduce the transfer learning method which is a fast and effective technique for utilizing similarity between different SWE trajectories while also taking into account their differences.

Echo State Network, Shallow Water Equations, Simulated Lasso, Multiscale Systems