Cross-Validation of Data-Driven Correction Reduced Order Modeling
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In this thesis, we develop a data-driven correction reduced order model (DDC-ROM) for numerical simulation of fluid flows. The general DDC-ROM involves two stages: (1) we apply ROM filtering (such as ROM projection) to the full order model (FOM) and construct the filtered ROM (F-ROM). (2) We use data-driven modeling to model the nonlinear interactions between resolved and unresolved modes, which solves the F-ROM's closure problem. In the DDC-ROM, a linear or quadratic ansatz is used in the data-driven modeling step. In this thesis, we propose a new cubic ansatz. To get the unknown coefficients in our ansatz, we solve an optimization problem that minimizes the difference between the FOM data and the ansatz. We test the new DDC-ROM in the numerical simulation of the one-dimensional Burgers equation with a small diffusion coefficient. Furthermore, we perform a cross-validation of the DDC-ROM to investigate whether it can be successful in computational settings that are different from the training regime.
General Audience Abstract
Practical engineering and scientific problems often require the repeated simulation of unsteady fluid flows. In these applications, the computational cost of high-fidelity full-order models can be prohibitively high. Reduced order models (ROMs) represent efficient alternatives to brute force computational approaches. In this thesis, we propose a data-driven correction ROM (DDC-ROM) in which available data and an optimization problem are used to model the nonlinear interactions between resolved and unresolved modes. In order to test the new DDC-ROM's predictability, we perform its cross-validation for the one-dimensional viscous Burgers equation and different training regimes.
- Masters Theses