From Filtering to Data-Driven Closures: Stabilization Strategies for Convection-dominated Flow Simulations

This PhD thesis investigates advanced stabilization strategies for the simulation of convection-dominated flows. The primary objective is to enhance the numerical stability and accuracy of classical methods through the integration of filtering and correction strategies. The first part of the thesis focuses on the stabilization of marginally-resolved simulations, where insufficient spatial resolution often leads to numerical instabilities, with the use of filtering strategies, in the so-called Evolve-Filter-Relax (EFR) framework. A comprehensive review of existing filtering strategies is presented, followed by the development of novel methodologies. These include an optimized EFR algorithm with parameter tuning over time, a reinforcement learning-based adaptive filtering strategy, and a novel data-driven energy-conserving EFR formulation. Together, these contributions demonstrate how adaptive and learnable filtering techniques can significantly improve stability and robustness. Motivated by the need to reduce the computational cost of high-fidelity simulations, the second part focuses on reduced order methods (ROMs) particularly in the Proper Orthogonal Decomposition (POD)-based framework. After reviewing the main stabilization techniques available in the literature, the thesis introduces novel data-driven filtering and correction methodologies to improve the performance of POD-based ROMs. These include replacing classical ROM filters with learned stabilization operators and introducing data-driven closure terms to recover the effects of the truncated POD modes. Furthermore, the proposed closure modeling strategies are extended to non-intrusive ROMs through machine-learning-based correction models operating in a purely data-driven setting. Overall, these methodologies significantly improve both the stability and the predictive accuracy of classical POD-based ROMs. While the second part remains inherently limited by the linear POD projection, the third part of the thesis explores the transition toward nonlinear reduced-order modeling. After reviewing the main nonlinear ROM methodologies available in the literature, the thesis investigates convolutional neural network architectures for the efficient compression of full-order operators, laying the foundations for more expressive and physics-based nonlinear ROMs. Overall, this work highlights the potential of combining stabilization strategies with machine learning and data-driven techniques to address key challenges in flow simulations, both on physical grids and in ROMs.