Lagrangian uncertainty quantification and information inequalities for stochastic flows

We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic Lagrangian (i.e., trajectory-based) predictions which are obtained from (Eulerian) vector fields generating the underlying dynamical system in a way which naturally applies in both deterministic and stochastic settings. This work is motivated by the desire to improve Lagrangian predictions in complex, multi-scale systems based on simplified, data-driven models. Here, discrepancies between probability measures $\mu$ and $\nu$ associated with the true dynamics and its approximation are quantified via so-called $\varphi$-divergencies, $\mathcal{D}_\varphi(\mu\|\nu)$, which are premetrics defined by a class of strictly convex functions $\varphi$. We derive general information bounds on the uncertainty in estimates, $\mathbb{E}^{\nu}[f]$, of `true' observables $\mathbb{E}^{\mu}[f]$ in terms of $\varphi$-divergencies; we then derive two distinct bounds on $\mathcal{D}_\varphi(\mu\|\nu)$ itself. First, an analytically tractable bound on $\mathcal{D}_\varphi(\mu\|\nu)$ is derived from differences between vector fields generating the true dynamics and its approximations. The second bound on $\mathcal{D}_\varphi(\mu\|\nu)$ is based on a difference of so-called finite-time divergence rate (FTDR) fields and it can be exploited within a computational framework to mitigate the error in Lagrangian predictions by tuning the fields of expansion rates obtained from simplified models. This new framework provides a systematic link between Eulerian (field-based) model error and the resulting uncertainty in Lagrangian (trajectory-based) predictions.