Probabilistic Learning on Manifolds for Statistical Surrogate Models in Uncertain Nano-to-Macro Stochastic Systems

This paper presents a probabilistic learning method on random manifolds for building and updating statistical surrogate models using small datasets. The approach accounts for various types of variables, including random, controlled, and latent, forming a random manifold that links quantities of interest to controlled variables. This paper consolidates the Probabilistic Learning on Manifolds (PLoM) methodology, previously dispersed across multiple publications, and demonstrates its effectiveness through three diverse applications: Molecular dynamics, nonlinear elasticity, and updating under-observed nonlinear dynamic system with incomplete data. These examples illustrate the robustness of the method in managing complex and uncertain problems.