Calibration-based ALE model order reduction for hyperbolic problems with self-similar travelling discontinuities (Q6635787)

The paper presents a novel framework for model order reduction (MOR) targeting hyperbolic problems characterised by self-similar travelling discontinuities.\N\NThe primary scientific problem addressed in this study pertains to the challenges faced by MOR techniques in efficiently approximating solutions of hyperbolic partial differential equations (PDEs), particularly those involving multiple travelling discontinuities, such as shocks and rarefaction waves. Traditional MOR approaches often struggle with such problems due to the slow decay of the Kolmogorov \( N \)-width, leading to a lack of reducibility for solutions exhibiting steep features.\N\NTo overcome these challenges, the authors propose a methodology leveraging an arbitrary Lagrangian-Eulerian (ALE) calibration approach. This involves transforming the solution manifold into a lower-dimensional representation by employing calibration maps derived through an optimisation process. Notably, the framework does not require explicit knowledge of the discontinuities' locations, thus avoiding the computationally intensive shock-tracking techniques typically employed. Instead, reference control points are used to perform the calibration, significantly reducing the computational overhead in the offline phase. The methodology integrates a non-intrusive approach for the online phase, utilising artificial neural networks (ANNs) to recover the coefficients of the reduced solution.\N\NThe manuscript describes rigorous validation through numerical experiments on benchmark problems, including the 1D Sod shock tube problem, the 2D double Mach reflection problem, and the triple point problem. In the Sod shock tube example, the proposed calibration technique successfully aligns features such as shocks and contact discontinuities across parameter spaces, enabling efficient MOR. The numerical results demonstrate significant improvements in the decay of the Kolmogorov \( N \)-width for the calibrated manifold compared to traditional approaches, highlighting the effectiveness of the proposed calibration in enhancing the efficiency of the MOR.\N\NThe findings of this study are significant for the field of computational mathematics, particularly in advancing the capabilities of MOR techniques for advection-dominated problems. By eliminating the need for explicit shock detection and leveraging machine learning models for reduced-order representations, the proposed framework offers a scalable and generalisable approach applicable to various hyperbolic PDEs. This contribution is expected to facilitate real-time computations and multi-query analyses in scientific and engineering applications, paving the way for further innovations in the domain of reduced-order modelling.