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We introduce a data-driven and physics-informed framework for propagatinguncertainty in stiff, multiscale random ordinary differential equations (RODEs)driven by correlated (colored) noise. Unlike systems subjected to Gaussianwhite noise, a deterministic equation for the joint probability densityfunction (PDF) of RODE state variables does not exist in closed form. Moreover,such an equation would require as many phase-space variables as there arestates in the RODE system. To alleviate this curse of dimensionality, weinstead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations forlow-dimensional observables/quantities of interest. The unclosed terms take theform of state-dependent conditional expectations, which are directly estimatedfrom data at sparse observation times. However, for systems exhibiting stiff,multiscale dynamics, data sparsity introduces regression discrepancies thatcompound during RoPDF evolution. This is overcome by introducing a kinetic-likedefect term to the RoPDF equation, which is learned by assimilating in sparse,low-fidelity RoPDF estimates. Two assimilation methods are considered, namelynudging and deep neural networks, which are successfully tested against MonteCarlo simulations.

Data-driven Closures & Assimilation for Stiff Multiscale Random Dynamics