Abstract
In the modern era, most systems of interest are too complex to be amenable to traditional deriv-
ations from first principles. As such, data-driven methods have experienced a surge in popularity,
and are revolutionizing the way we model, predict and control dynamical systems Many of these
data-driven methods employ neural networks within their architecture, which undergo a compu-
tationally expensive training phase to optimize a model over a fixed dataset. Dynamical systems
however are often continuously generating new data in the form of a data stream, and to maintain
the most accurate and up-to-date models, this new data must be incorporated in some way into
the model. Regularly retraining the model is often computationally infeasible, hence there is a need
for techniques capable of efficiently modelling dynamical systems directly from data streams.
Existing methods for modelling linear dynamical systems from data streams often adapt the
existing data-driven modelling technique dynamic mode decomposition (DMD), and adapt it for
the streaming data case. These methods often suffer from either developing models that are
insensitive to information in new measurements and unable to track a changing system, or being
overly responsive to new data and hence highly unstable in the presence of measurement noise. We
show how DMD can be combined with the data assimilation technique ensemble Kalman filtering
(EnKF), in what we refer to as the DMDEnKF, to effectively update the current state and model
estimates of a system as new data becomes available. We demonstrate that this approach is able
to track time varying dynamical systems in synthetic examples, and when applied to real world
seasonal influenza-like illness data from the USA Centers for Disease Control and Prevention, find
the DMDEnKF’s performance in short-term forecasting to be comparable to the best mechanistic
models in the ILINet competition.
Many dynamical systems of interest are nonlinear, so when modelling these systems DMD
based frameworks have limited applicability, as they produce linear models. Data-driven modelling
techniques exist for nonlinear dynamical systems, however they are limited by poor dimensional
scaling, high sensitivity to abstract user-specified parameters, or being difficult to train. One such
method is the Koopman autoencoder, which generates a low-dimensional latent space and model
of a nonlinear system directly from data. We propose small alterations to Koopman autoencoders
to make them easier to train, then show how their reduced order latent space and model of the
system can be efficiently updated as new data becomes available using the EnKF, in a technique
we call the KAE EnKF. We demonstrate this approach is able to effectively track and forecast
time-varying, nonlinear dynamical systems in synthetic examples. We then apply the KAE EnKF
to a real video of a pendulum, and achieve a significant improvement over current state-of-the-art
methods, in generating effective latent space reconstructions, accurate short-term forecasts, and
efficient adaptations to externally forced changes to the pendulum’s frequency.