Journal
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Volume 380, Issue -, Pages -Publisher
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2021.113777
Keywords
Probabilistic learning; PLoM; Partial differential equations; Small dataset; Machine learning; Uncertainty quantification
Funding
- Defense Advanced Research Projects Agency (DARPA) [HR00111990032]
- FASTMATH, a SciDAC Institute (Scientific Discovery through Advanced Computing) - U.S. Department of Energy (DOE), Office of Science, United States
- Advanced Scientific Computing Research (ASCR), United States
Ask authors/readers for more resources
This paper introduces a novel probabilistic learning algorithm that can solve a wide range of nonlinear stochastic boundary value problems depending on vector-valued random control parameters. The algorithm allows for generating a large number of learned realizations of stochastic processes and their corresponding random control parameters, and generates these realizations by minimizing the vector-valued random residual of the PDE.
A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier-Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint. (C) 2021 ElsevierB.V. All rights reserved.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available