Data-driven discovery of model error in chaotic systems by integrating Bayesian sparse regression and data assimilation

We propose the application of a Bayesian machine learning technique to discover a closed-form representation of the model error, i.e. the symbolic representation of the difference between the true dynamics of a system and its corresponding model. The technique incorporates a data assimilation step and therefore is capable of handling noisy observations. The models of complex physical phenomena such as fluid flows, climate, and weather often contain missing representations of certain hard-to-model processes, which leads to prediction inaccuracies. An increasing abundance of data from observations provides the opportunity to discover these missing physics, and therefore to improve the predictive capabilities of numerical, purely data-driven, or hybrid models with minimal changes. Accordingly, we propose the use of relevant vector machines (RVM), an inherently sparsity promoting Bayesian model, in conjunction with a stochastic ensemble Kalman filter (EnKF) to discover the closed form of model error from noisy observations. We demonstrate the robustness of our technique in the presence of different magnitudes of noise in a well-known challenging task in system identification, i.e. Kuramoto–Sivashinsky (KS) equation in a chaotic regime.