Monday  |  Salon 9  |  03:00 PM–03:20 PM

#16812–Universal Upper Estimate for Prediction Errors Under Moderate Model Uncertainty

We present a method of sensitivity analysis for general dynamical systems subjected to deterministic or stochastic modeling uncertainty. Using the properties of the unperturbed, idealized dynamics, we derive a universal bound for the leading-order prediction error. Specifically, our estimates give upper bounds on the leading order trajectory-uncertainty arising along model trajectories, solely as functions of the invariants of the known Cauchy-Green strain tensor of the idealized model. Our bounds turn out to be optimal, which means that they cannot be improved for general systems. This bound motivates the definition of a quantity called Model Sensitivity, which is a scalar depending on the initial condition and time and quantifies the general susceptibility of the system to modeling errors. Using nonlinear numerical models of various complexities, we demonstrate that the Model Sensitivity provides both a global view over the phase space of the dynamical system and, in some situations, a localized, time-dependent predictor of uncertainties along trajectories. This is reflected by the fact that the mean-squared trajectory uncertainty qualitatively follows the leading-order bound for surprisingly long time intervals. We also find that the phase-space structure of the Model Sensitivity (MS) is related but not identical to another commonly used metric for sensitivity, the Finite-Time Lyapunov Exponents (FTLE). We formulate conditions under which the FTLE field's robust features are also expected to be seen in the MS field.

Bálint Kaszás   ETH Zurich
George Haller   ETH Zurich