DDPS | Model reduction via optimization of projection operators and reduced-order dynamics

DDPS Talk date: July 19th, 2024

Speaker: Alberto Padovan (UIUC, https://www.linkedin.com/in/alberto-padovan-7b0416272/)

Description: Computing reduced-order models using non-intrusive methods is particularly attractive for systems that are simulated using black-box solvers. However, obtaining accurate data-driven models can be challenging, especially if the underlying systems exhibit large-amplitude transient growth. Although these systems may evolve near a low-dimensional subspace that can be easily identified using standard techniques such as Proper Orthogonal Decomposition (POD), computing accurate models often requires projecting the state onto this subspace via a non-orthogonal projection. While appropriate oblique projection operators can be computed using intrusive techniques that leverage the form of the underlying governing equations, purely data-driven methods currently tend to achieve dimensionality reduction via orthogonal projections, and this can lead to models with poor predictive accuracy. We address this issue by introducing a non-intrusive framework designed to simultaneously identify oblique projection operators and reduced-order dynamics directly from data. In particular, given training trajectories and assuming reduced-order dynamics of polynomial form, we fit a reduced-order model by solving an optimization problem over the product manifold of a Grassmann manifold, a Stiefel manifold, and several linear spaces (as many as the tensors that define the low-order dynamics). Furthermore, we show that the gradient of the cost function with respect to the optimization parameters can be conveniently written in closed form, so that there is no need for automatic differentiation. This formulation is compared with state-of-the-art methods on three examples: a three-dimensional system of ordinary differential equations, the complex Ginzburg-Landau (CGL) equation, and a two-dimensional lid-driven cavity flow at Reynolds number Re = 8300.

Bio: Alberto is a Postdoctoral Research Associate in the Department of Aerospace Engineering at the University of Illinois Urbana-Champaign. He obtained his PhD from Princeton University in August 2022, and his research interests lie at the intersection of fluid mechanics, dynamical systems and control theory. Alberto’s current work focuses on the development of data-driven methods for model reduction of fluid flows, as well as on the analysis and control of supersonic and hypersonic wall-bounded flows.

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IM release number is: LLNL-VIDEO-867054

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