High-dimensional, nonlinear, multi-scale phenomena, such as turbulence or the spread of infectious diseases, are ubiquitous; however, we still lack a good understanding of these as analytically tractable models remain an exception. The lack of simple equations and unprecedented amount of available high-fidelity data are leading to a paradigm shift in how we interact with complex nonlinear systems. Leading approaches stem from data-driven methods which have the potential to discover new mechanisms, models and control laws and are driven by the tremendous advances in computing power, new sensors and infrastructures, and advanced algorithms in machine learning.
In this talk, I will discuss recent advances in data-driven, equation-free architectures leveraging sparsity-promoting techniques for the modeling and control of dynamical systems. One direction is connected to Koopman operator theory, which has emerged as a principled framework to obtain linear embeddings of nonlinear dynamics, enabling the estimation, prediction and control of strongly nonlinear systems using standard linear techniques. A data-driven architecture is presented for the identification of Koopman eigenfunctions using sparse regression and polynomial expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. In addition, I will discuss work related to statistical modeling in fluids and how to exploit sparsity in dynamical systems for sensing. The presented work is demonstrated on Hamiltonian systems and different high-dimensional systems from fluids.