How Large-Time Behavior of Turbulent Flows Determine the Amount of Data Needed for Their Prediction

Turbulence is often claimed to be "the last unsolved problem of classical mechanics," realized in the evolution of certain dynamical systems, most notably those for fluid dynamics.  In general, these systems behave nicely in that their large time behavior is finite dimensional and possibly even smooth, but the set of all possible solutions is chaotic.  This leads to incredible difficulties in predictive modeling.  For predictive modeling, one generally uses data assimilation (DA) or machine learning to course-correct models with real-world data.  Often overlooked in the literature is the importance of dynamical behavior; machine learning models or data assimilation techniques are regularly applied (and sometimes expected to work) without regard to the physics of the system they are being applied to (indeed, the algorithms are written to be agnostic to the dynamics in general).  In this talk, I will present various research papers exposing different perspectives of how accurate prediction from data is, provably, entirely dependent on the underlying dynamics of the system under consideration.  This talk presents multiple research papers which are joint work with many collaborators from Caltech, Hunter College, BYU, USC, and UVa.