  In many areas of science and engineering, discovering the governing
differential equations from the noisy experimental data is an essential
challenge. It is also a critical step in understanding the physical phenomena
and prediction of the future behaviors of the systems. However, in many cases,
it is expensive or time-consuming to collect experimental data. This article
provides an active learning approach to estimate the unknown differential
equations accurately with reduced experimental data size. We propose an
adaptive design criterion combining the D-optimality and the maximin
space-filling criterion. In contrast to active learning for other regression
models, the D-optimality here requires the unknown solution of the differential
equations and derivatives of the solution. We estimate the Gaussian process
(GP) regression models from the available experimental data and use them as the
surrogates of these unknown solution functions. The derivatives of the
estimated GP models are derived and used to substitute the derivatives of the
solution. Variable selection-based regression methods are used to learn the
differential equations from the experimental data. Through multiple case
studies, we demonstrate the proposed approach outperforms the D-optimality and
the maximin space-filling design alone in terms of model accuracy and data
economy.
