Reduced-order modeling with reconstruction-informed projections

Reduced-order models (ROMs) are commonly employed to address the large computational cost of simulations for high-dimensional, dynamical systems such as those found in combustion. Two key components of building these ROMs include an effective dimension reduction or projection of the high-dimensional system onto a low-dimensional manifold, alongside an accurate method for reconstructing quantities of interest (QoIs) over the projection parameters. A QoI-based approach to projection has been recently introduced, combining these aspects of building ROMs into a single training that learns the projection parameters based on nonlinear reconstruction errors of QoIs. A key advantage to this approach is the ability to inform the projection with reconstruction errors of QoIs that are functions of the projection itself, such as projected source terms influencing evolution over the low-dimensional manifold. We expand on the QoI-based projection method and implement Partition of Unity Network (POUnet) models for reconstruction, which are able to reach high levels of accuracy with small memory footprints by combining classification with localized polynomial regression. Furthermore, we find that sign changes in the projected source terms can negatively impact evolution over the low-dimensional manifold. Therefore, we also include a penalty function favoring positivity in projected source terms of the QoI-based projection method. These modeling improvements are demonstrated in one- and two-dimensional direct numerical simulations (DNS) of a non-premixed laminar flame.