Non-linear Manifold Reduced-Order Models with Convolutional Autoencoders and Reduced Over-Collocation Method

Non-affine parametric dependencies, nonlinearities and advection-dominated regimes of the model of interest can result in a slow Kolmogorov n-width decay, which precludes the realization of efficient reduced-order models based on linear subspace approximations. Among the possible solutions, there are purely data-driven methods that leverage autoencoders and their variants to learn a latent representation of the dynamical system, and then evolve it in time with another architecture. Despite their success in many applications where standard linear techniques fail, more has to be done to increase the interpretability of the results, especially outside the training range and not in regimes characterized by an abundance of data. Not to mention that none of the knowledge on the physics of the model is exploited during the predictive phase. In order to overcome these weaknesses, we implement the non-linear manifold method introduced by Lee and Carlberg (J Comput Phys 404:108973, 2020) with hyper-reduction achieved through reduced over-collocation and teacher-student training of a reduced decoder. We test the methodology on a 2d non-linear conservation law and a 2d shallow water models, and compare the results obtained with a purely data-driven method for which the dynamics is evolved in time with a long-short term memory network.

Automated summary

Non-affine parametric dependencies, nonlinearities, and advection-dominated regimes can hinder the development of efficient reduced-order models based on linear subspace approximations. Data-driven methods utilizing autoencoders and their variants have shown promise, but there is a need for increased interpretability, especially in regions with limited data and outside the training range. Additionally, exploiting knowledge of the model's physics during the predictive phase is important. To address these challenges, we implement the non-linear manifold method introduced by Lee and Carlberg (J Comput Phys 404:108973, 2020) and combine it with hyper-reduction achieved through reduced over-collocation and teacher-student training of a reduced decoder. We evaluate the methodology on a 2D non-linear conservation law and a 2D shallow water model, comparing it to a purely data-driven approach using a long-short term memory network for time evolution.