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.

翻译：我们提出了一种基于数据驱动和物理信息的框架，用于传播受相关（有色）噪声驱动的刚性多尺度随机常微分方程中的不确定性。与受高斯白噪声影响的系统不同，随机常微分方程状态变量的联合概率密度函数不存在封闭形式的确定性方程。此外，这样的方程所需的相空间变量数量等同于随机常微分方程系统状态数。为缓解这一维数灾难，我们推导出针对低维可观测变量/感兴趣量的精确但未封闭的降阶概率密度方程。未封闭项表现为状态依赖的条件期望，可直接从稀疏观测时间点的数据中估计。然而，对于具有刚性多尺度动态的系统，数据稀疏性引入的回归偏差会在降阶概率密度方程的演化过程中累积。我们通过引入类似动能的缺陷项来克服这一问题，该项通过同化稀疏低精度的降阶概率密度估计值进行学习。我们考虑了两种同化方法，即 nudging 和深度神经网络，并在蒙特卡洛模拟中成功验证了其有效性。