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Models of many engineering and natural systems are imperfect. The discrepancybetween the mathematical representations of a true physical system and itsimperfect model is called the model error. These model errors can lead tosubstantial differences between the numerical solutions of the model and thestate of the system, particularly in those involving nonlinear, multi-scalephenomena. Thus, there is increasing interest in reducing model errors,particularly by leveraging the rapidly growing observational data to understandtheir physics and sources. Here, we introduce a framework named MEDIDA: ModelError Discovery with Interpretability and Data Assimilation. MEDIDA onlyrequires a working numerical solver of the model and a small number ofnoise-free or noisy sporadic observations of the system. In MEDIDA, first themodel error is estimated from differences between the observed states andmodel-predicted states (the latter are obtained from a number of one-time-stepnumerical integrations from the previous observed states). If observations arenoisy, a data assimilation (DA) technique such as ensemble Kalman filter (EnKF)is employed to provide the analysis state of the system, which is then used toestimate the model error. Finally, an equation-discovery technique, here therelevance vector machine (RVM), a sparsity-promoting Bayesian method, is usedto identify an interpretable, parsimonious, and closed-form representation ofthe model error. Using the chaotic Kuramoto-Sivashinsky (KS) system as the testcase, we demonstrate the excellent performance of MEDIDA in discoveringdifferent types of structural/parametric model errors, representing differenttypes of missing physics, using noise-free and noisy observations.

chaos

Discovery of interpretable structural model errors by combining Bayesian  sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test  case