Eric Parish

Ayoub Gouasmi, Eric Parish, Karthik Duraisamy

Reduced Order Models (ROMs) of complex, nonlinear dynamical systems often require closure, which is the process of representing the contribution of the unresolved physics on the resolved physics. The Mori-Zwanzig (M-Z) procedure allows one to write down formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the past history of the resolved variables, and a "noise" term. While the M-Z procedure does not directly reduce the complexity of the original system, these equations can serve as a mathematically consistent starting point to develop closures based on approximations of the memory. In this scenario, a priori knowledge of the memory kernel, which is not explicitly known for nonlinear systems, is of paramount importance to assess the validity of a memory approximation. Unraveling the memory kernel requires the determination of the orthogonal dynamics which is a projected high-dimensional partial differential equation that is not tractable in general. A method to estimate the memory kernel a priori, using full-order solution snapshots, is proposed. The main idea is to solve a pseudo orthogonal dynamics equation, that has a convenient Liouville form, instead of the original one. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics operator is a composition operator, akin to semi-groups of Liouville operators, for one observable. The method is exact in the linear case where the kernel is known explicitly. Results for under-resolved simulations of the Burgers and Kuramoto-Sivashinsky equations demonstrate that the proposed technique can accurately reconstruct the transfer of information between the resolved and unresolved dynamics through memory, and provide valuable information about the kernel.

Jorio Cocola, John Tencer, Francesco Rizzi et al.

Projection-based model order reduction on nonlinear manifolds has been recently proposed for problems with slowly decaying Kolmogorov n-width such as advection-dominated ones. These methods often use neural networks for manifold learning and showcase improved accuracy over traditional linear subspace-reduced order models. A disadvantage of the previously proposed methods is the potential high computational costs of training the networks on high-fidelity solution snapshots. In this work, we propose and analyze a novel method that overcomes this disadvantage by training a neural network only on subsampled versions of the high-fidelity solution snapshots. This method coupled with collocation-based hyper-reduction and Gappy-POD allows for efficient and accurate surrogate models. We demonstrate the validity of our approach on a 2d Burgers problem.