Parametric model order reduction techniques often struggle to capture transport-dominated phenomena because the Kolmogorov n-width decays slowly. To tackle this issue,
we introduce a data-driven method that combines sPOD with deep learning. First, we use sPOD to create a high-fidelity, low-dimensional model of the system, which then feeds into a deep learning network to predict how the system behaves under different parameters.
One challenge with sPOD is that it needs the shifts to be known before-hand. To address this, we also propose a neural network approach that learns both the shifts and the co-moving low-rank fields at the same time by using two specialized sub-networks. This is a first step towards an automated sPOD method, and we are also exploring how PINN-style learning can help improve the separation of the transport field and the detection of shifts.
Finally, we show how reduced order models, which are often used as simpler surrogates for full order models in control or optimization tasks, can be applied in an optimal control setting using a basic linear transport example.