Bagging, optimized dynamic mode decomposition for robust, stable forecasting with spatial and temporal uncertainty quantification

Abstract

Dynamic mode decomposition (DMD) provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of temporal, or spatio-temporal, data. A variety of regression techniques have been developed for producing the linear model approximation whose solutions are exponentials in time. For spatio-temporal data, DMD provides low-rank and interpretable models in the form of dominant modal structures along with their exponential/oscillatory behaviour in time. The majority of DMD algorithms, however, are prone to bias errors from noisy measurements of the dynamics, leading to poor model fits and unstable forecasting capabilities. The optimized DMD algorithm minimizes the model bias with a variable projection optimization, thus leading to stabilized forecasting capabilities. Here, the optimized DMD algorithm is improved by using statistical bagging methods whereby a single set of snapshots is used to produce an ensemble of optimized DMD models. The outputs of these models are averaged to produce a bagging, optimized dynamic mode decomposition (BOP-DMD). BOP-DMD improves performance by stabilizing and cross-validating the DMD model by ensembling; it also robustifies the model and provides both spatial and temporal uncertainty quantification (UQ). Thus, unlike currently available DMD algorithms, BOP-DMD provides a stable and robust model for probabilistic , or Bayesian, forecasting with comprehensive UQ metrics. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

Transition to chaos and modal structure of magnetized Taylor–Couette flow

“…This is not a concerning issue here, since we used DMD not for reduced-order modelling of the system, but as a diagnostic tool for its dynamical behaviour. The future work will include relating the temporal evolution of the modes into a nonlinear model, together with improving robustness of the presented DMD method by taking into account flow symmetries [ 20 , 21 ], or harnessing statistical properties of the flow [ 22 ]. Such a model could provide a quantitative description of nonlinear interactions accompanying the transition to MRI turbulence.…”

Characterization of Acoustic Emissions From Analogue Rocks Using Sparse Regression‐DMDc

“…Currently there are several emerging methods that allow for a broader viewpoint of building models directly from noisy data, recent innovations in DMD, for instance, have shown that statistical bagging methods can greatly increase the discovery of robust, accurate and stable linear models (Askham & Kutz, 2018; Sashidhar & Kutz, 2021). This has motivated the use of ensembling and bagging for building nonlinear, parsimonious dynamic models (Fasel et al., 2022; Hirsh et al., 2021).…”

“…There are a number of variants for computing A (Kutz et al., 2016), with the exact DMD simply positing A = X ′ X † where † denotes the Moore‐Penrose pseudo‐inverse. However, the optimized‐DMD (Askham & Kutz, 2018) (opt‐DMD) and bagging‐optimized DMD (Sashidhar & Kutz, 2021) (BOP‐DMD) provide algorithms that provide substantial performance gains when considering noisy data. However, such improved algorithms have yet to be incorporated with control architetures.…”

Data-driven prediction in dynamical systems: recent developments

“…The majority of classical DMD algorithms, however, are prone to bias errors from noisy measurements of the dynamics, leading to poor model fits and unstable forecasting capabilities. Sashidhar & Kutz [ 187 ] introduce an optimized DMD method, called bagging optimized dynamic mode decomposition (BOP-DMD), by using statistical bagging methods that improves the performance of DMD methods. Unlike currently available DMD algorithms, BOP-DMD provides a stable and robust model for probabilistic or Bayesian forecasting with comprehensive uncertainty quantification metrics.…”