Current Research

Physics-Informed Data-Driven Modeling

Machine learning is gradually becoming an appealing alternative to first-principles modeling in physical sciences and engineering. For low-dimensional systems, the governing physical laws can often be inferred directly from the data without relying on any prior domain knowledge. For high-dimensional systems purely data-driven approaches become intractable. We are developing hybrid approaches which incorporate elements of first-principles, physically-informed approaches into machine learning framework (SPIDER) capable of identifying the governing equations for spatially extended systems using experimental or numerical data. As an example, SPIDER has been used to infer a complete mathematical model of an experimental active nematic system (some results can be found here) which generated new physical insight. We are currently working on applying this framework in several new areas:
• Validating and troubleshooting numerical codes
• Subgrid-scale modeling of multi-scale phenomena
• Continuum modeling of systems of discrete interacting units

Transition to Turbulence in Fluid Flows

Fluids tend to undergo a transition from laminar to turbulent flow as the rate of shear increases. Although both transition mechanisms and the statistics of turbulent motion can differ widely, turbulent flows in any geometry share some key common features. While the flow is generally very complicated and seemingly disorganized, it occasionally organizes itself to resemble, for a short time, one or another recurrent structure. These structures have been identified as exact regular (steady, time-periodic, and so on) solutions of the Navier-Stokes equation. Our research is focused on testing the hypothesis that these unstable regular solutions can be used to construct a reduced-order model of weakly turbulent flows. Once constructed, this low-dimensional description will be used to develop a nonlinear flow control algorithms capable of relaminarizing fully developed turbulence. This problem is currently being explored in the context of several canonical geometries: 2D Kolmogorov flow, Taylor-Couette flow, and 2D/3D channel flow.

Arrhythimic Dynamics of Cardiac Tissue

Our research uses the tools and techniques of dynamical systems to study cardiac arrhythmias ranging from mild tachycardia to lethal fibrillation, with the ultimate goal of developing effective and reliable defibrillation methods. Numerical simulations and experimental studies have identified fibrillation with the breakup of spiral waves of excitation in the heart tissue. However, neither approach has produced a clear picture of what the mechanism responsible for initiating and sustaining fibrillation is. Furthermore, it is well known that both the normal beating and fibrillation are natural behaviors even in healthy hearts. Fibrillation can be provoked and stopped by an external perturbation. However the threshold for fibrillatory dynamics is much lower in disease-ridden hearts. The transition mechanism therefore is not related to the classical supercritical instabilities typical of pattern-forming systems but rather involves a subcritical instability akin to the transition to turbulence in fluid flows.