Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems

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 introduce 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 dual 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. Apart from this, the algorithm has also been tested for a case where the data has been taken from an experimental setup (Silver box dataset). The results produced further showcase the efficacy of the proposed framework.