  A framework for creating and updating digital twins for dynamical systems
from a library of physics-based functions is proposed. The sparse Bayesian
machine learning is used to update and derive an interpretable expression for
the digital twin. Two approaches for updating the digital twin are proposed.
The first approach makes use of both the input and output information from a
dynamical system, whereas the second approach utilizes output-only observations
to update the digital twin. Both methods use a library of candidate functions
representing certain physics to infer new perturbation terms in the existing
digital twin model. In both cases, the resulting expressions of updated digital
twins are identical, and in addition, the epistemic uncertainties are
quantified. In the first approach, the regression problem is derived from a
state-space model, whereas in the latter case, the output-only information is
treated as a stochastic process. The concepts of It\^o calculus and
Kramers-Moyal expansion are being utilized to derive the regression equation.
The performance of the proposed approaches is demonstrated using highly
nonlinear dynamical systems such as the crack-degradation problem. Numerical
results demonstrated in this paper almost exactly identify the correct
perturbation terms along with their associated parameters in the dynamical
system. The probabilistic nature of the proposed approach also helps in
quantifying the uncertainties associated with updated models. The proposed
approaches provide an exact and explainable description of the perturbations in
digital twin models, which can be directly used for better cyber-physical
integration, long-term future predictions, degradation monitoring, and
model-agnostic control.
