Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints

Abstract

In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology in a parallel computing framework. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.

arXiv comment30 pages, 6 figures

References in corpus (19)

• Deep Learning in Neural Networks: An Overview
• DGM: A deep learning algorithm for solving partial differential equations
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• A Physics Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies Based on DNS Data
• Machine Learning of Linear Differential Equations using Gaussian Processes
• A Comprehensive Physics-Informed Machine Learning Framework for Predictive Turbulence Modeling
• Physics-Informed Machine Learning Approach for Augmenting Turbulence Models: A Comprehensive Framework
• Bayesian Inference with Optimal Maps
• Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark
• A modern maximum-likelihood theory for high-dimensional logistic regression
• Kullback-Leibler upper confidence bounds for optimal sequential allocation
• Sparse Regression Learning by Aggregation and Langevin Monte-Carlo
• Data-driven probability concentration and sampling on manifold
• Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold
• Reduced Wiener Chaos representation of random fields via basis adaptation and projection
• On the Brittleness of Bayesian Inference
• Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets
• Random fields representations for stochastic elliptic boundary value problems and statistical inverse problems
• Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements