EnKode: Active Learning of Unknown Flows with Koopman Operators

TL;DR

This letter proposes EnKode, an active sampling approach based on Koopman Operator theory and ensemble methods that can build high quality flow models and effectively estimate model uncertainty and provides comparable or better estimates of unsampled flow regions than Gaussian Process Regression models with hyperparameter optimization.

Abstract

In this letter, we address the task of adaptive sampling to model vector fields. When modeling environmental phenomena with a robot, gathering high resolution information can be resource intensive. Actively gathering data and modeling flows with the data is a more efficient alternative. However, in such scenarios, data is often sparse and thus requires flow modeling techniques that are effective at capturing the relevant dynamical features of the flow to ensure high prediction accuracy of the resulting models. To accomplish this effectively, regions with high informative value must be identified. We propose EnKode, an active sampling approach based on Koopman Operator theory and ensemble methods that can build high quality flow models and effectively estimate model uncertainty. For modeling complex flows, EnKode provides comparable or better estimates of unsampled flow regions than Gaussian Process Regression models with hyperparameter optimization. Additionally, our active sensing scheme provides more accurate flow estimates than comparable strategies that rely on uniform sampling. We evaluate EnKode using three common benchmarking systems: the Bickley Jet, Lid-Driven Cavity flow with an obstacle, and real ocean currents from the National Oceanic and Atmospheric Administration (NOAA).

Figures (7)

• Figure 1: EnKode modeling and active sensing scheme. Flow measurements are acquired by a robot, and fed into EnKode, which combines ensemble methods and Koopman operator theory to model unknown nonlinear ordinary differential equations with model uncertainty quantification. Measurements are encoded via lifting functions into a space where dynamics are linear, propagated forward in time by the estimated Koopman operator, and reconstructed to produce a flow estimate and uncertainty map. The next best sampling location is that with the greatest epistemic uncertainty, a measure of information, providing a feedback loop for the next best sampling location.
• Figure 2: Pixel-wise EPE after actively modeling the flow of the Bickley Jet, for $N_{total} = 36$. Overall EnKode error is lower than that of GP-32 w/ opt. There are localized regions with errors for the GP that do not appear in the EnKode estimate.
• Figure 3: Errors when actively modeling Bickley Jet. ME values closer to $0$ are desired, while CS closer to $1$ are desired. For $N \geq 10$, ME is lower and CS is higher when actively sampling with EnKode.
• Figure 4: Top. Pixel-wise EPE for Lid-Driven Cavity with Obstacle. Bottom. Corresponding streamlines. When actively sampling, EnKode captures both the bulk flow dynamics and small vortices, while the GP is able to capture the bulk flow only.
• Figure 5: Errors for modeling the Lid-Driven Cavity flow with obstacle over $N \leq 36$ samples. If active sampling is used, EnKode performs better than the GP baseline.