  With the advent of modern data collection and storage technologies,
data-driven approaches have been developed for discovering the governing
partial differential equations (PDE) of physical problems. However, in the
extant works the model parameters in the equations are either assumed to be
known or have a linear dependency. Therefore, most of the realistic physical
processes cannot be identified with the current data-driven PDE discovery
approaches. In this study, an innovative framework is developed that combines
data-driven and data-assimilation methods for simultaneously identifying
physical processes and inferring model parameters. Spatiotemporal measurement
data are first divided into a training data set and a testing data set. Using
the training data set, a data-driven method is developed to learn the governing
equation of the considered physical problem by identifying the occurred (or
dominated) processes and selecting the proper empirical model. Through
introducing a prediction error of the learned governing equation for the
testing data set, a data-assimilation method is devised to estimate the
uncertain model parameters of the selected empirical model. For the contaminant
transport problem investigated, the results demonstrate that the proposed
method can adequately identify the considered physical processes via
concurrently discovering the corresponding governing equations and inferring
uncertain parameters of nonlinear models, even in the presence of measurement
errors. This work helps to broaden the applicable area of the research of data
driven discovery of governing equations of physical problems.

Machine Learning;Machine Learning