Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

Abstract

This paper deals with the taking into account a given target set of realizations as constraints in the Kullback–Leibler divergence minimum principle (KLDMP). We present a novel probabilistic learning algorithm that makes it possible to use the KLDMP when the constraints are not defined by a target set of statistical moments for the quantity of interest (QoI) of an uncertain/stochastic computational model, but are defined by a target set of realizations for the QoI for which the statistical moments associated with these realizations are not or cannot be estimated. The method consists in defining a functional constraint, as the equality of the Fourier transforms of the posterior probability measure and the target probability measure, and in constructing a finite representation of the weak formulation of this functional constraint. The proposed approach allows for estimating the posterior probability measure of the QoI (unsupervised case) or of the posterior joint probability measure of the QoI with the control parameter (supervised case). The existence and the uniqueness of the posterior probability measure is analyzed for the two cases. The numerical aspects are detailed in order to facilitate the implementation of the proposed method. The presented application in high dimension demonstrates the efficiency and the robustness of the proposed algorithm.

Appendix A. Generation of the training set, target set, and numerical values of the parameters

The training set $D_d = \{ {\varvec{x}}^1,\ldots , {\varvec{x}}^{N_d}\}$ with is made up of $N_d$ independent realizations of random variable ${\varvec{X}}=({\varvec{Q}},{\varvec{W}})$, which are generated by using a stochastic computational model corresponding to the finite element discretization of a stochastic elliptic boundary value problem for which $n_x=430{,}098$, $n_q=10{,}098$, and $n_w= 420{,}000$. The target set $D_{\textrm{targ}}= \{ {\varvec{q}}^1_{\textrm{targ}},\ldots , {\varvec{q}}^{N_r}_{\textrm{targ}}\}$ is generated using the stochastic computational model with another values of the parameters (see “Appendix A.3”).

1.1 Appendix A.1. Definition of the stochastic boundary value problem

Let $\Omega = ]\,0\, , 1\,[\, \times \, ]\,0\, , 0.2\,[\,\times \,]\,0 \, , 0.1\,[ \, m^3$ be the bounded open set of ${\mathbb {R}}^3$, with generic point ${\varvec{\omega }}= (\omega _1,\omega _2,\omega _3)$, and with boundary $\partial \Omega =\Gamma _0 \cup \Gamma _1\cup \Gamma _2$ in which $\Gamma _0 =\{\omega _1=1\, , \, 0\le \omega _2 \le 0.2 \, , \, 0 \le \omega _3 \le 0.1 \}$, $\Gamma _1 =\{\omega _1=0\, , \, 0\le \omega _2 \le 0.2 \, , \, 0 \le \omega _3 \le 0.1 \}$, and $\Gamma _2 =\partial \Omega \backslash \{\Gamma _0\cup \Gamma _1 \}$. Let be ${\overline{\Omega }}=\Omega \cup \partial \Omega$. The outward unit normal to $\partial \Omega$ is denoted by . We use the usual convention of summation on repeated Latin indices. Domain $\Omega$ is occupied by a heterogeneous and anisotropic elastic random medium for which the elastic properties are defined by the fourth-order tensor-valued non-Gaussian random field . Let be the ${\mathbb {R}}^3$-valued displacement random field defined in $\Omega$. A Dirichlet condition is given on $\Gamma _0$ while a Neumann condition is given on $\Gamma _1\cup \Gamma _2$. The stochastic boundary value problem is written, for $k=1,2,3$ and almost surely, as

(A.1)

(A.2)

(A.3)

(A.4)

in which the stress tensor is related to the strain tensor by by the constitutive equation, . For $k=1,2,3$, the applied stresses $p_k$ on $\Gamma _1$ are defined as follows:

$p_1 = 0$ on $\Gamma _1$, except:

$\quad p_1 = -1.8\times 10^8\, N/m^2$ for ${\varvec{\omega }}\in \{\omega _1=0\, , \, 0\le \omega _2\le 0.02 \, , \, 0\le \omega _3 \le 0.1\}$.

$\quad p_1 = +9.0\times 10^7\, N/m^2$ for ${\varvec{\omega }}\in \{\omega _1=0\, , \, 0.18\le \omega _2\le 0.2 \, , \, 0\le \omega _3 \le 0.1\}$.

$p_2 = 0$ on $\Gamma _1$, except:

$\quad p_2 = +1.0\times 10^7\, N/m^2$ for ${\varvec{\omega }}\in \{\omega _1=0\, , \, \{0\le \omega _2\le 0.02\}\cup \{0.18\le \omega _2\le 0.20\} \, , \, 0\le \omega _3 \le 0.02\}$.

$\quad p_2 = -1.5\times 10^7\, N/m^2$ for ${\varvec{\omega }}\in \{\omega _1=0\, , \, \{0\le \omega _2\le 0.02\}\cup \{0.18\le \omega _2\le 0.20\} \, , \, 0.08\le \omega _3 \le 0.1\}$.

$p_3 = 0$ on $\Gamma _1$, except:

$\quad p_3 = -2.40\times 10^7\, N/m^2$ for ${\varvec{\omega }}\in \{\omega _1=0\, , \, 0\le \omega _2\le 0.02 \, , \, 0\le \omega _3 \le 0.1\}$.

$\quad p_3 = +2.64\times 10^7\, N/m^2$ for ${\varvec{\omega }}\in \{\omega _1=0\, , \, 0.18\le \omega _2\le 0.2 \, , \, 0\le \omega _3 \le 0.1\}$.

Using the matrix representation in Voigt notation, the random elasticity field is rewritten, for k, m, n, and q in $\{1,2,3\}$, as with ${{\textbf{i}}}=(k,m)$ with $1\le k\le m\le 3$ and ${{\textbf{j}}}=(n,q)$ with $1\le n \le q \le 3$ in which indices ${{\textbf{i}}}$ and ${{\textbf{j}}}$ belong to $\{1,\ldots , 6\}$. The -valued random field $\{ [{\varvec{A}}({\varvec{\omega }})] ,{\varvec{\omega }}\in \Omega \}$ is a non-Gaussian, second order, and statistically homogeneous. Its mean function is the given ${\varvec{\omega }}$-independent matrix corresponding to a homogeneous isotropic elastic material whose Young modulus is $10^{10}\, N/m^2$ and Poisson coefficient 0.15 (note that the fluctuations around the mean are those of a heterogeneous anisotropic elastic material). The non-Gaussian -valued random field $\{ [{\varvec{A}}({\varvec{\omega }})]\, ,{\varvec{\omega }}\in \Omega \}$ is constructed using the stochastic model (Soize 2006, 2008b, 2017) of random elasticity fields for heterogeneous anisotropic elastic media that are isotropic in statistical mean and exhibit anisotropic statistical fluctuations, for which the parameterization consists of spatial-correlation lengths and of a positive-definite lower bound. The random field $\{[{\varvec{A}}({\varvec{\omega }})],{\varvec{\omega }}\in \Omega \}$ is written as,

(A.5)

in which is the upper triangular $(6\times 6)$ real matrix such that , where $\epsilon$ is a given positive number (which can be chosen arbitrarily small), and where $\{ [{\varvec{G}}({\varvec{\omega }})],{\varvec{\omega }}\in {\mathbb {R}}^3\}$ is a -valued random field (by construction), defined on , indexed by ${\mathbb {R}}^3$. Then $[{\varvec{G}}]$ is homogeneous, mean-square continuous, and such that $E\{[{\varvec{G}}({\varvec{\omega }}))]\} = [I_6]$ for all ${\varvec{\omega }}\in {\mathbb {R}}^3$. Note that the lower bound $\epsilon \,[\,{\underline{{\varvec{A}}}}\, ]/(1+\epsilon )$ used in Eq. (A.5) could be replaced by a more general lower bound $[A_b]$ in as proposed in Guilleminot and Soize (2013) and Soize (2017). For all ${\varvec{\omega }}$ fixed in ${\mathbb {R}}^3$, the -valued random variable $[{\varvec{G}}({\varvec{\omega }})]$ has been constructed by using the Maximum Entropy Principle under the following available information, $E\{[{\varvec{G}}({\varvec{\omega }})]\} = [I_6]$ and $E\{ \log (\det [{\varvec{G}}({\varvec{\omega }})] ) \} = b_G$ with $\vert b_G \vert \, < +\infty$, which has been introduced in order that the random matrix $[{\varvec{G}}({\varvec{\omega }})]^{-1}$ (that exists almost surely) be such that $E\{\Vert [{\varvec{G}}({\varvec{\omega }})]^{-1}\Vert ^2\} \le$ $E\{\Vert [{\varvec{G}}({\varvec{\omega }})]^{-1}\Vert _F^2\} < +\infty$. In this construction, for all ${\varvec{\omega }}$ fixed in ${\mathbb {R}}^3$, is a -valued nonlinear function [g(.)] of $6\times (6+1)/2 = 21$ independent normalized Gaussian real-valued random variables denoted by and such that and . The spatial correlation structure of random field $\{[{\varvec{G}}({\varvec{\omega }})],$ ${\varvec{\omega }}\in {\mathbb {R}}^3\}$ is introduced by considering 21 independent real-valued random fields for $1\le m \le n \le 6$, corresponding to 21 independent copies of a unique normalized Gaussian homogeneous mean-square continuous real-valued random field whose normalized spectral measure is given and has a support that is controlled by three spatial correlation lengths $L_{c1} = L_{c2} = L_{c3} = 0.4$. Note that this Gaussian field can be replaced by a non-Gaussian field for taking into account uncertainties in the spectral measure (Soize 2021). The constant $b_G$ is eliminated in favor of a hyperparameter $\delta _G > 0$, which allows for controlling the level of statistical fluctuations of $[{\varvec{G}}({\varvec{\omega }})]$, defined by $\delta _G =(E\{\Vert [{\varvec{G}}({\varvec{\omega }})] - [I_6]\Vert _F^2\} / 6)^{1/2}$, which is independent of ${\varvec{\omega }}$ and such that $\delta _G= 0.6$.

1.2 Appendix A.2. Stochastic computational model for generating the training set $D_d$ and observed quantities of interest

The stochastic boundary value problem defined by Eqs. (A.1) to (A.4) is discretized by the finite element method. Domain $\Omega$ is meshed with $50\times 10\times 5 = 2500$ finite elements using 8-nodes finite elements. There are 3366 nodes and 10, 098 dofs (degrees of freedom). The displacements are locked at all the 66 nodes belonging to surface $\Gamma _0$ and therefore, there are 198 zero Dirichlet conditions. There are 8 integration points in each finite element. Consequently, there are $N_p= 20{,}000$ integration points ${\varvec{\omega }}^1,\ldots , {\varvec{\omega }}^{N_p}$. The -valued random variable ${\varvec{W}}$ is generated as follows. For all $p=1,\ldots , N_p$, let in which $\log _M$ is the logarithm of positive-definite matrices. The -valued random variable ${\varvec{W}}$ is then defined as the vector that is the reshaping of the upper triangular part of the $N_p$ matrices $\{\, [{\varvec{G}}_p^{\textrm{log}}], p=1,\ldots , N_p\}$.We then have $n_w = 21\times N_p = 420{,}000$. The finite element discretization of random field is the -valued random variable ${\varvec{Q}}$ with $n_q= 10{,}098$. Consequently ${\varvec{X}}=({\varvec{Q}},{\varvec{W}})$ is a random variable with values in with $n_x=n_q+n_w = 430{,}098$. The stochastic computational model is then represented by a stochastic linear matrix equation that is solved by using the Monte Carlo numerical simulation method yielding the training set $D_d = \{ {\varvec{x}}^1,\ldots , {\varvec{x}}^{N_d}\}$ in which is a realization of random variable ${\varvec{X}}=({\varvec{Q}},{\varvec{W}})$, the computed realizations being independent. For studying the convergence properties, the considered values of $N_d$ are $N_d\in \{100, 200, 300, 400 \}$.

The components of the quantity of interest ${\varvec{Q}}$, which will be observed for presenting the results, are the 3 components denoted by $Q_{{\textrm{obs}},1}$, $Q_{{\textrm{obs}},2}$, and $Q_{{\textrm{obs}},3}$ that correspond to the 3 dofs along directions $\omega _1$, $\omega _2$, and $\omega _3$ of the finite element node of coordinates (0, 0, 0.1) (located at top corner in which the displacements are significant and result from tension, torsion, and two bendings contributions).

1.3 Appendix A.3. Target set of realizations

The target set $D_{\textrm{targ}}= \{{\varvec{q}}_{\textrm{targ}}^1,\ldots {\varvec{q}}_{\textrm{targ}}^{N_r}\}$ is generated using the stochastic boundary value problem defined in Section Appendix A.1 for which the elasticity matrix $[{\underline{{\varvec{A}}}}^{\textrm{targ}}]$ is the one of a homogeneous and isotropic elastic material with a Young modulus $9\times 10^9\, \hbox {N/m}^2$ and a Poisson coefficient $\nu =0.15$. The level of statistical fluctuations of the random field $\{{\varvec{G}}^{\textrm{targ}}({\varvec{\omega }}),{\varvec{\omega }}\in {\mathbb {R}}^3\}$ is $\delta _G^{\textrm{targ}}= 0.3$. In order to analyze the convergence with respect to $N_r$, we have considered, in consistency with the values of $N_d$, the intervals $N_r \in [50\, , N_{\textrm{targ}}]$ with $N_{\textrm{targ}}\in \{100,200,300,400\}$.