Abstract: Complex nonlinear dynamic phenomena, e.g., spatiotemporal chaos, often require many degrees of freedoms to model, making them challenging to simulate and analyze. Reduced models that use only those dynamical variables that are of direct scientific relevance are thus useful for reducing computational cost and potentially offering insights into dynamical mechanisms. However, without sharp timescale separation, reduced models can exhibit memory effects and non-Markovian, history-dependent behavior, and a challenge for model reduction is to capture such effects efficiently. The focus of this talk is a discrete-time version of the Mori-Zwanzig (MZ) projection operator formalism from nonequilibrium statistical mechanics that makes use of Koopman operator ideas from ergodic theory. I will show that the MZ formalism provides a useful general framework within which to construct reduced models for chaotic and stochastic dynamical systems without scale separation. As an example, I will sketch a derivation of the NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous inputs) representation of stochastic processes, widely used for data driven modeling of time series data, and show how this can be applied to prototypical models of chaotic and stochastic behavior in spatially extended systems. I will conclude with a discussion of open questions and future directions.

Bio: Kevin K Lin received undergraduate degrees in Computer Science and in Mathematics from MIT, an MEng in Computer Science (also from MIT) under the direction of Gerald Jay Sussman, and a PhD in Mathematics from UC Berkeley in 2003 under the direction of Alexandre Chorin. He was an NSF Postdoctoral Fellow at the Courant Institute advised by Lai-Sang Young and has been on the Mathematics faculty at the University of Arizona (UA) since 2007. At UA, he is a member of the graduate interdisciplinary programs in Applied Mathematics, Statistics and Data Science, Neuroscience, and Cognitive Science. His work has been supported by the National Science Foundation, the Fannie and John Hertz Foundation, and the Simons Foundation. His research interests center around dynamical systems, computing, and applications, especially to computational neuroscience.