A new methodology for rigorous Bayesian learning of high-dimensional stochastic dynamical models is developed. The methodology performs parallelized computation of marginal likelihoods for multiple candidate models, integrating over all state variable and parameter values, and enabling a principled Bayesian update of model distributions. This is accomplished by leveraging the dynamically orthogonal (DO) evolution equations for uncertainty prediction in a dynamic stochastic subspace and the Gaussian Mixture Model-DO filter for inference of nonlinear state variables and parameters, using reduced-dimension state augmentation to accommodate models featuring uncertain parameters. Overall, the joint Bayesian inference of the state, model equations, geometry, boundary conditions, and initial conditions is performed. Results are exemplified using two high-dimensional, nonlinear simulated fluid and ocean systems. For the first, limited measurements of fluid flow downstream of an obstacle are used to perform joint inference of the obstacle’s shape, the Reynolds number, and the $\mathcal{O}\left(10^5\right)$ fluid velocity state variables. For the second, limited measurements of the concentration of a microorganism advected by an uncertain flow are used to perform joint inference of the microorganism’s reaction equation and the $\mathcal{O}\left(10^5\right)$ microorganism concentration and ocean velocity state variables. When the observations are sufficiently informative about the learning objectives, we find that our posterior model probabilities correctly identify either the true model or the most plausible models, even in cases where a human would be challenged to do the same.

开发了一种用于高维随机动力学模型的严格贝叶斯学习的新方法。该方法为多个候选模型执行边际似然的并行计算，整合所有状态变量和参数值，并实现模型分布的原则性贝叶斯更新。这是通过利用动态正交 (DO) 演化方程在动态随机子空间中进行不确定性预测并利用高斯混合模型-DO 滤波器来推断非线性状态变量和参数，使用降维状态增强来适应具有不确定参数的模型来实现的. 总体而言，执行状态、模型方程、几何、边界条件和初始条件的联合贝叶斯推理。使用两个高维非线性模拟流体和海洋系统举例说明了结果。首先，障碍物下游的流体流动的有限测量用于对障碍物的形状、雷诺数和$\mathcal{哦}\left(10^5\right)$流体速度状态变量。对于第二个，由不确定流平流的微生物浓度的有限测量用于联合推断微生物的反应方程和$\mathcal{哦}\left(10^5\right)$微生物浓度和海洋速度状态变量。当观察结果足以提供关于学习目标的信息时，我们发现我们的后验模型概率可以正确识别真实模型或最合理的模型，即使在人类面临同样挑战的情况下也是如此。