Speaker: Alberto Padovan, University of Illinois at Urbana-Champaign

Time : 10:00 - 11.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy, room 128-129.

Abstract:

Developing accurate reduced-order models (ROMs) for high-dimensional physical systems (e.g., fluids, solids, systems in thermochemical kinetics, etc.) is essential to accelerate simulations and enable engineering tasks like design and optimization. In this talk, we emphasize the importance of oblique projections for the development of ROMs with long-time predictive accuracy, and we present recent advances in their computation using both intrusive and non-intrusive methods.

Several systems in engineering, including transport-dominated fluid flows, exhibit high sensitivity to low-energy coordinates. These coordinates are typically truncated in ROMs obtained with orthogonal projections designed to capture the high-energy features of the system. As a consequence of this truncation, the resulting ROMs can fail to faithfully reproduce the dynamics of the original system. By contrast, oblique projections can be used to compute ROMs that capture the energetic features of the flow while also accounting for the low-energy coordinates that are necessary to accurately describe the dynamics of the full-order model. In this presentation, we present two novel formulations to compute oblique-projection-based (or Petrov-Galerkin) models for nonlinear systems. The first one is similar in spirit to the well-known balanced truncation formulation for linear systems, and it identifies a quasi-optimal oblique projector by balancing the state and gradient covariance matrices associated with the flow of the underlying full-order model. The second method is a refinement of the latter, where oblique projections for Petrov-Galerkin model reduction are further optimized against high-fidelity data to reduce forecasting error. While these two methods can work very well and outperform existing model reduction formulation, they are highly intrusive and require direct access to the underlying governing equations. They may therefore be inapplicable to systems that are simulated using black-box solvers (e.g., commercial software or legacy codes) whose source code might be proprietary or very difficult to modify. To address this issue, we introduce a third (non-intrusive) formulation, which we call “NiTROM: Non-intrusive Trajectory-based optimization of ROMs.” Specifically, given high-fidelity data from a black-box solver, NiTROM computes a ROM by simultaneously seeking the optimal projection and latent-space dynamics that minimize forecasting error. This formulation can achieve predictive accuracy comparable to that of the first two (intrusive) Petrov-Galekin methods, and it has been shown that it can significanly outperform state-of-the-art non-intrusive frameworks like Operator Inference. All the methods discussed in this talk are demonstrated on several systems, including incompressible fluid flows with 𝑂(105) states.