  For real-life nonlinear systems, the exact form of nonlinearity is often not
known and the known governing equations are often based on certain assumptions
and approximations. Such representation introduced model-form error into the
system. In this paper, we propose a novel gray-box modeling approach that not
only identifies the model-form error but also utilizes it to improve the
predictive capability of the known but approximate governing equation. The
primary idea is to treat the unknown model-form error as a residual force and
estimate it using duel Bayesian filter based joint input-state estimation
algorithms. For improving the predictive capability of the underlying physics,
we first use machine learning algorithm to learn a mapping between the
estimated state and the input (model-form error) and then introduce it into the
governing equation as an additional term. This helps in improving the
predictive capability of the governing physics and allows the model to
generalize to unseen environment. Although in theory, any machine learning
algorithm can be used within the proposed framework, we use Gaussian process in
this work. To test the performance of proposed framework, case studies
discussing four different dynamical systems are discussed; results for which
indicate that the framework is applicable to a wide variety of systems and can
produce reliable estimates of original system's states.
