Manifold Learning Analysis and Bayesian Latent-Observational Feature Prediction of a Large Eddy Simulated Turbulence Field

A latent-observational space analytical formulism is applied to a sub-grid modeled turbulent kinetic energy (tke) field emanating from ocean turbulence large eddy simulation (LES) data containing Langmuir cells but no breaking waves. The purpose of the analysis is to illustrate how machine learning modeling can be used to understand the probabilistic structure of observational space and how the allied latent space can be related statistically to it for the purpose of data generation. The Peter and Clark (PC)-algorithm-based Bayesian belief network (BBN) edge-nodal structure for the observational-space tke subdomains demonstrates a distinctive nonlocal connectivity pattern when the multidimensional scaling graph layout is invoked. When the Chow-Liu algorithm is used, tree-based connectivity in the network is revealed. In particular, a dominant parental root node occupies the far upper left region in the observational-space domain with many edge connections flowing toward the right. Hidden Markov model (HMM) parameter estimation, applied to the maximum and minimum values taken from the mean tke feature matrix and to generative topographic mapping (GTM)based latent space for the same tke feature nodes, enables estimation of the latent-space state transition matrix and tke latent-observational space emission matrices. The latent-space state transition matrix demonstrates how many columns of latent space, associated with different root-mean-square tke values, possess a high probability of transitioning to other distinct latent-space areas. The tke latent-observational space emission matrix provides spatial subdomain locations most strongly tied to specific vertical columns of GTM-based latent space. The maximum value-based emission matrix shows very high probabilities for single columns on the latent-space perimeter being associated with maximum values occurring at specific observational-space subdomain locations. These observational-space nodal locations have strong statistical linkages to other nodes when the Chow-Liu-algorithm-based BBN is invoked, suggesting this model to be physically appropriate to the high energy turbulent flow physics. Bayesian and manifold learning processing methods provide a way for understanding the spatial structure of LES-derived tke features and how latent space can provide a constraint for discerning optimal linkages between spatially separate observational-space subdomains.