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

arXiv: https://arxiv.org/abs/2202.05112 · HTML: https://mlsanity.com/paper/2202.05112/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 151 references that are in the map.
- First posted on arXiv 10 Feb 2022.

## Builds on (most cited first, 14 of 14)

- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018)
- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [Machine Learning-augmented Predictive Modeling of Turbulent Separated Flows over Airfoils](https://mlsanity.com/paper/1608.03990/) (2016)
- [Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets](https://mlsanity.com/paper/2010.14324/) (2020)
- [Maximum Likelihood Estimation for Learning Populations of Parameters](https://mlsanity.com/paper/1902.04553/) (2019)
- [Probabilistic Learning on Manifolds (PLoM) with Partition](https://mlsanity.com/paper/2102.10894/) (2021)
- [Goal-oriented optimal approximations of Bayesian linear inverse problems](https://mlsanity.com/paper/1607.01881/) (2016)
- [Sampling of Bayesian posteriors with a non-Gaussian probabilistic learning on manifolds from a small dataset](https://mlsanity.com/paper/1910.12717/) (2019)
- [A Bayesian Perspective of Statistical Machine Learning for Big Data](https://mlsanity.com/paper/1811.04788/) (2018)
- [Reduced Wiener Chaos representation of random fields via basis adaptation and projection](https://mlsanity.com/paper/1603.04803/) (2016)

## Built on by (most cited first, 2 of 2)

- [Transient anisotropic kernel for probabilistic learning on manifolds](https://mlsanity.com/paper/2407.21435/) (2024)
- [Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures](https://mlsanity.com/paper/2205.03078/) (2022)

## Abstract (arXiv)

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.