A Local ROM for Rayleigh–Bénard Bifurcation Problems

This work presents a local reduced-order method for computing bifurcation diagrams in 2D Rayleigh–Bénard convection problems. The proposed method is based on Proper Orthogonal Decomposition, and employs a reduced-order study of the regularity of solutions to detect new solution branches within the bifurcation diagram. The locality of the method is achieved through k-means clustering, and the selection of local problems in the online stage is done in the solution space. All hyperparameters of the reduced method, such as the number of clusters or the number of POD modes, are estimated by deterministic criteria. Furthermore, the offline–online splitting strategy for online calculations is explicitly outlined. The method is applied to a single-parameter problem and a two-parameter problem, showing its ability to rapidly compute bifurcation diagrams with small errors. Compared to a standard approach that samples each branch separately, the local approach produces more accurate results in less computational time.