Non-intrusive model reduction of advection-dominated hyperbolic problems using neural network shift augmented manifold transformations

Advection-dominated problems are predominantly noticed in nature, engineering systems, and various industrial processes. Traditional linear compression methods, such as proper orthogonal decomposition (POD) and reduced basis (RB) methods are ill-suited for these problems, due to slow Kolmogorov $n$-width decay. This results in inefficient and inaccurate reduced order models (ROMs). There are few non-linear approaches to accelerate the Kolmogorov $n$-width decay. In this work, we use a neural network shift augmented transformation technique that employs automatic shift detection. This approach leverages a deep-learning framework to derive a parameter-dependent mapping between the original manifold $\mathcal{M}$ and the transformed manifold $\tilde{\mathcal{M}}$. We apply a linear compression method to obtain a low-dimensional linear approximation subspace of the transformed manifold $\tilde{\mathcal{M}}$. Furthermore, we construct non-intrusive reduced order models on the resulting transformed linear approximation subspace and employ automatic shift detection for predictions in the online stage. We propose a complete framework, the neural network shift-augmented proper orthogonal decomposition-based reduced order model (NNsPOD-ROM) algorithm, comprising both offline and online stages for model reduction of advection-dominated problems. We test our proposed methodology on numerous experiments to evaluate its performance on the 1D linear advection equation, a higher order method benchmark case - the 2D isentropic convective vortex, and 2D two-phase flow.