Systems of Gaussian process models for directed chains of solvers

Abstract

The simulation of complex multi-physics phenomena often relies on a System of Solvers (SoS), which we define here as a set of interdependent solvers where the output of an upstream solver is the input of downstream solvers. Constructing a surrogate model of a SoS presents a clear interest when multiple evaluations of the system are needed, for instance to perform uncertainty quantification and global sensitivity analyses, the resolution of optimization or control problems, and generally any task based on fast query evaluations. In this work, we develop an original mathematical framework, based on Gaussian Process (GP) models, to construct a global surrogate model of the directed SoS, (i.e., only featuring one-way dependencies between solvers). The two central ideas of the proposed approach are, first, to determine a local GP model for each solver constituting the SoS and, second, to define the prediction as the composition of the individual GP models constituting a system of GP models (SoGP). We further propose different adaptive sampling strategies for the construction of the SoGP. These strategies use the decomposition of the SoGP prediction variance into individual contributions of the constitutive GP models and on extensions to SoGP of the Maximum Mean Square Predictive Error criterion. We finally assess the performance of the SoGP framework on several SoS involving different numbers of solvers and structures of input dependencies. The results show that the SoGP framework is very flexible and can handle different types of SoS, with a significantly reduced construction cost (measured by the number of training samples) compared to constructing a unique GP model of the SoS. , systems of solvers have received interest from the community trying to develop efficient UQ methods for SoS. In [11], the authors proposed a method based on importance sampling to decouple the uncertainty propagation process of individual solvers in order to gain flexibility. Other recent works focused on adapting Global Polynomial Chaos (gPC) based methods to SoS, with the challenge of deriving efficient quadrature rules on intermediate inputs with unknown distributions. Using the structure of SoS, the authors of [12] proposed a method for propagating uncertainty in a composite function by adapting the quadrature rule of intermediate inputs in the SoS, thus limiting the number of quadrature points compared to a global black box approach. This work used the recursive formula for orthogonal polynomials and Lanczos algorithms. The same authors generalized this idea in [13] to a full SoS. Their approach relies on Galerkin projection methods at intermediate layers of the SoS. By solving an optimization problem, they proposed a quadrature rule for latent variables, regularized in order to promote sparsity in the weights, thus reducing the number of quadrature points. In [14], the authors tackled the problem of strongly coupled systems. Their main idea is that the dimension of the coupling variables and the amount of information transferred from ...

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