A strongly nonlinear system is a chaotic dynamical system which exhibits energy transfer between various scales in time and space. The well-known examples of this behavior commonly appear in engineering and geophysical turbulence. In this project, we work on capturing relatively low-dimensional but statistically accurate models of these systems with the aid of data . The end goal is to use these models, instead of very costly physics-based models, to characterize and predict extreme events (e.g. heat waves), study the statistical response to external output (e.g. climate trends), and synthesize long-time control strategies (e.g. mean drag reduction).
Many physical, chemical and biological systems are described at a microscopic level (e.g. molecular reaction, response of neurons to stimula and vehicle platoons) but it is suspected that they may admit evolution equations at much larger length scales, e.g. in the form of PDEs. Discovering such coarse-grained evolution laws can lead to huge savings in computation-intensive tasks like prediction, design optimization and control. There are two challenges in machine learning these coarse-grained evolution laws from data: 1- the need for ample trainging data from microscopic simulations, and 2- ambiguity in the choice of coarse-grained dependent variable (e.g. density, pair probability etc.) in those equations. We combine modern tools of applied mathematics; such as equation-free numerics, optimal transport of probabilities, and manifold learning; to address these challenges.
Why do we hear so much about data? One reason is that we have a lot of it, but also in the context of dynamical systems, data is becoming instrumental in dealing with problems that have large dimensions and contain uncertainy. The traditional approach in dynamical systems theory has been centered around geometric analysis in the state space, but a lot of systems that we are interested in right now have either too many degrees of freedom to model directly (e.g. climate, turbulent flows and chemical kinetics) or lack reliable models for state-space analysis (e.g. biological and social networks). Data helps us detect and exploit low-dimensional structures in the former group, or find surrogate but amenable models for the latter. Two sets of techniques have contributed the most to data-driven analysis and control in the recent years: 1-operator-theoretic methods and 2-machine learning. Our goal is to push the boundaries of application for these methods, or invent others, that makes them applicable to problems ranging from turbulence and climate to cell motility and health monitoring.
The goal of this project is to devise and analyze numerical algorithms that enable data-driven analysis and control of dynamical systems. For example, Dynamic Mode Decomposition (DMD) is a linear data decomposition algorithm that has become the primary tool for computing the Koopman eigenfunctions (i.e. linearly evolving coordinates) in simple nonlinear systems. In previous works, we have established rigorous results on the connection between this algorithm [1,2], and some other signal processing methods , with operator-theoretic viewpoint for nonlinear systems. We are interested in improving the performance of numerical toolbox for more efficient and robust data-driven modeling, for example by including optimal sampling of state and inputs.
Mixing is a crucial part of many processes related to fluids like movement of oil slicks in the ocean, processing of polymers and metals, and biodmedical testing. Despite its prevalence, the process of mixing in flows, even in the absence of diffusion and particle-size effects, is not fully understood. Most of rigorous analysis in the field of advective mixing comes from the theory of chaotic advection which treats the fluid flow as a dynamical system with trajectories that describe the paths of tracers. A strong theme of our work is to understand mixing by combining the fluid mechanical apsect (i.e. Navier-Stokes dynamics) with the methods of chaotic advection. For example, this research has led to extension of classical Prandtl-Batchelor theory to unsteady 2D flows  which predicts that at high Reynolds the rotational structures in the flow resemble rigid body rotation. Interstingly, resonance arguments from chaotic advection show that this constant-vorticity condition implies that mixing in the core of those flows would be weaker than wall-adjacent areas, which is radically different from mixing in low-Reynolds flows .
On a more applied side, we collaborated with a group in Croatia to design an efficient algorithm for search on the ocean surface. The new algorithm , which combines ideas from statistical mechanics and ocean mixing, has shown great promise in a computational replication of search for the missing Malaysian Airliner MH370. Another exciting collaboration was formed with scientists in Marine Science Institute to devise a framework for prediction of oil spill movement on the ocean . This problem was motivated by the 2015 Refugio oil spill in northern Santa Barbara county.