From intrusive to non-intrusive model reduction : structure preservation and exact operator inference

Computational efficiency is of paramount importance in complex engineering tasks such as optimization, data assimilation, uncertainty quantification and the development of digital twins. All these tasks require multiple simulations with similar settings. While classical numerical methods such as Finite Elements have primarily been developed to run single simulations, data-driven methods are being developed to exploit the similarity among such multiple simulations. One of these data-driven methods is projection-based model reduction. Like classical numerical methods, this method derives discrete models from first principles. However, while classical numerical methods result in generic discretizations, projection-based model reduction leverages data to construct more efficient, problem-specific discretizations. This thesis comprises three advances of projection-based model reduction. The first advance is the application to the incompressible Navier-Stokes equations with time-dependent boundary conditions. This application results in a reduced-order model (ROM) that does not require the computation of the pressure to simulate the velocity and preserves the structure of the kinetic energy evolution. It is shown that this velocity-only ROM is equivalent to a velocity-pressure ROM, i.e., a ROM that simulates both velocity and pressure. The advantages of the velocity-only ROM are better stability and computational efficiency. The second advance is the proposition of a method that enables the construction of projection-based ROMs non-intrusively, i.e., without access to the corresponding full-order model. Such access is often not possible with software used in industry, so operator inference has been introduced for non-intrusive ROM construction. However, operator inference reconstructs the corresponding intrusive ROM only if sufficiently informative snapshot data is available. The novel method generates such data and thereby guarantees the exact reconstruction. This exact operator inference is optimal in the sense that the amount of generated data to achieve exact reconstruction is minimal. The third advance is the application of exact operator inference to parametric models. Next to the exact reconstruction with minimal data, this application allows to decouple the parametric operator inference minimization problem. The resulting decoupled problems are embarrassingly parallel, numerically more stable and can be solved more efficiently. The main result of this thesis is the proposed exact operator inference that enables the exact non-intrusive reconstruction of intrusive ROMs. This method can be applied to many scientific and engineering problems, including the incompressible Navier-Stokes equations as studied in this thesis.

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