Discovery of interpretable structural model errors by combining Bayesian sparse regression and data-assimilation*

Loss in accuracy of models originates from inaccuracies of  

i) estimation of the state of the system, 

ii) models of physical processes, i.e., model error.

In data-assimilation, measurements and observations are often used to correct the deviation of the predicted state from the observed state of the system. Hence, the prediction horizon is increased.

Although such approach reduces the effect of the model error by accounting for the deviation, it does not provide any knowledge regarding the source of the model error. Recent efforts towards increasing both amount and frequency of observations of the system, specifically the Earth system, provides an opportunity to  leverage the data to identify a closed-form representation of the model error.

We have scaled our proposed framework, MEDIDA (Model Error Discovery with Interpretability and Data-Assimilation), to high dimensional systems.

Here, Ensemble Kalman filter, a traditional data-assimilation technique, is combined with a Bayesian sparse identification of nonlinear dynamics. The EnKF provides a noise-reduced estimation of the state of the system, analysis state, which is then used to form a regression problem to identify the closed form of the model error. Moreover, an artificial neural-network with random Fourier feature is used to estimate the full state of the system given sparse in space observations. The method is then applied to one and two dimensional systems, e.g., chaotic Kuramoto–Sivashinsky and quasi-geostrophic turbulence.