Joint probabilistic modelling and sampling from small data via probabilistic learning on manifolds and decoupled multi-probability density evolution method

Abstract

Computational models are of utmost importance in various aspects of structural design and optimization, uncertainty quantification, risk assessment, and other engineering fields. Among numerous critical issues for model-based uncertainty quantification, the issue of joint probabilistic modelling of dependent model parameters under the condition of small data is investigated in this work. In contrast to most viewpoints, which see model parameters by observation or experiment as mere data, we regard them as stochastic responses that are realized through implicit (and possibly complicated) physical processes with respect to underlying random variables. Based on this thought, a novel approach for estimating the joint probabilistic model of $n$-dimensional dependent model parameters is proposed in two stages: Firstly, the probabilistic learning on manifolds (PLoM) is adopted to generate ample “virtual” realizations of model parameters that are statistically consistent with the original small data. This aims to construct an approximately dependent probability structure of model parameters; Secondly, the decoupled multi-probability density evolution method (PDEM) is employed to calculate the joint probabilistic model from a perspective of uncertainty propagation, where the original small data are equipped with assigned probabilities that are calculated from the dependent probability structure via PLoM. Moreover, taking the advantages of the proposed method, a low-dimensional data storing and a novel acceptance–rejection sampling technique are proposed, which is particularly convenient for the case that $n\ge 3$. A benchmark case is studied to illustrate and verify the proposed method, which is found to be robust to complex engineering data. Three applications, including double doughnut configuration-type’s data, mechanical parameters of concrete, and location and time parameters of typhoon genesis, are presented to demonstrate the new capacities of the proposed method.