  Learning time-dependent partial differential equations (PDEs) that govern
evolutionary observations is one of the core challenges for data-driven
inference in many fields. In this work, we propose to capture the essential
dynamics of numerically challenging PDEs arising in multiscale modeling and
simulation -- kinetic equations. These equations are usually nonlocal and
contain scales/parameters that vary by several orders of magnitude. We
introduce an efficient framework, Densely Connected Recurrent Neural Networks
(DC-RNNs), by incorporating a multiscale ansatz and high-order
implicit-explicit (IMEX) schemes into RNN structure design to identify analytic
representations of multiscale and nonlocal PDEs from discrete-time observations
generated from heterogeneous experiments. If present in the observed data, our
DC-RNN can capture transport operators, nonlocal projection or collision
operators, macroscopic diffusion limit, and other dynamics. We provide
numerical results to demonstrate the advantage of our proposed framework and
compare it with existing methods.
