CONMAN

Data-driven modeling has been having a transformative impact in many scientific fields. Of particular interest here are control-oriented techniques such as POD, ERA, or DMD. Despite their widespread usage, such techniques cannot easily provide a faithful representation of the system under scrutiny when the latter is parameterized. The aim of the present project is to tackle this issue. By reformulating these different techniques in the framework of reduced-rank regression, an optimal data-driven SVD factorization of the underlying operator can be obtained. Building on recent works about interpolation on matrix manifold, this factorization can be interpolated based on the Stiefel manifold, i.e. the manifold of orthogonal matrices. Using these new mathematical tools, the aim of the present project is to explore our ability to obtain parameterized data-driven models with a particular emphasis on closed-loop feedback control in fluid dynamics and to quantify the accuracy and robustness of these models. In order to illustrate our methodology, two classical problems from flow control (cylinder and shear-driven cavity flow) will be considered. In both cases, different flavours of reduced-order models will be considered (e.g. ERA, DMD, BPOD, etc) and parametric dependencies built into them. Because using the Stiefel manifold equips us with a natural notion of distance between two modal decompositions, efficient sampling strategies of the parameter space can be devised. Given data-driven models obtained at different points in the parameter space, an approximate model can be obtained for a new query point with minimal computational complexity using such interpolation schemes. Given the ubiquity of matrix decompositions in engineering sciences, the methodology proposed in this project is expected to have far-reaching applications, not only in fluid dynamics but also in data science and machine learning in general.