  Measured data from a dynamical system can be assimilated into a predictive
model by means of Kalman filters. Nonlinear extensions of the Kalman filter,
such as the Extended Kalman Filter (EKF), are required to enable the joint
estimation of (possibly nonlinear) system dynamics and of input parameters. To
construct the evolution model used in the prediction phase of the EKF, we
propose to rely on the Sparse Identification of Nonlinear Dynamics (SINDy).
SINDy enables to identify the evolution model directly from preliminary
acquired data, thus avoiding possible bias due to wrong assumptions and
incorrect modelling of the system dynamics. Moreover, the numerical integration
of a SINDy model leads to great computational savings compared to alternate
strategies based on, e.g., finite elements. Last, SINDy allows an immediate
definition of the Jacobian matrices required by the EKF to identify system
dynamics and properties, a derivation that is usually extremely involved with
physical models. As a result, combining the EKF with SINDy provides a
data-driven computationally efficient, easy-to-apply approach for the
identification of nonlinear systems, capable of robust operation even outside
the range of training of SINDy. To demonstrate the potential of the approach,
we address the identification of a linear non-autonomous system consisting of a
shear building model excited by real seismograms, and the identification of a
partially observed nonlinear system. The challenge arising from the use of
SINDy when the system state is not entirely accessible has been relieved by
means of time-delay embedding. The great accuracy and the small uncertainty
associated with the state identification, where the state has been augmented to
include system properties, underscores the great potential of the proposed
strategy, paving the way for the setting of predictive digital twins in
different fields.
