  Models of many engineering and natural systems are imperfect. The discrepancy
between the mathematical representations of a true physical system and its
imperfect model is called the model error. These model errors can lead to
substantial differences between the numerical solutions of the model and the
state of the system, particularly in those involving nonlinear, multi-scale
phenomena. Thus, there is increasing interest in reducing model errors,
particularly by leveraging the rapidly growing observational data to understand
their physics and sources. Here, we introduce a framework named MEDIDA: Model
Error Discovery with Interpretability and Data Assimilation. MEDIDA only
requires a working numerical solver of the model and a small number of
noise-free or noisy sporadic observations of the system. In MEDIDA, first the
model error is estimated from differences between the observed states and
model-predicted states (the latter are obtained from a number of one-time-step
numerical integrations from the previous observed states). If observations are
noisy, 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 to
estimate the model error. Finally, an equation-discovery technique, here the
relevance vector machine (RVM), a sparsity-promoting Bayesian method, is used
to identify an interpretable, parsimonious, and closed-form representation of
the model error. Using the chaotic Kuramoto-Sivashinsky (KS) system as the test
case, we demonstrate the excellent performance of MEDIDA in discovering
different types of structural/parametric model errors, representing different
types of missing physics, using noise-free and noisy observations.
