  We introduce a data-driven and physics-informed framework for propagating
uncertainty in stiff, multiscale random ordinary differential equations (RODEs)
driven by correlated (colored) noise. Unlike systems subjected to Gaussian
white noise, a deterministic equation for the joint probability density
function (PDF) of RODE state variables does not exist in closed form. Moreover,
such an equation would require as many phase-space variables as there are
states in the RODE system. To alleviate this curse of dimensionality, we
instead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations for
low-dimensional observables/quantities of interest. The unclosed terms take the
form of state-dependent conditional expectations, which are directly estimated
from data at sparse observation times. However, for systems exhibiting stiff,
multiscale dynamics, data sparsity introduces regression discrepancies that
compound during RoPDF evolution. This is overcome by introducing a kinetic-like
defect term to the RoPDF equation, which is learned by assimilating in sparse,
low-fidelity RoPDF estimates. Two assimilation methods are considered, namely
nudging and deep neural networks, which are successfully tested against Monte
Carlo simulations.
