Automated Manifold Learning for Reduced Order Modeling

The problem of identifying geometric structure in data is a cornerstone of (unsupervised) learning. As a result, Geometric Representation Learning has been widely applied across scientific and engineering domains. In this work, we investigate the use of Geometric Representation Learning for the data-driven discovery of system dynamics from spatial-temporal data. We propose to encode similarity structure in such data in a spatial-temporal proximity graph, to which we apply a range of classical and deep learning-based manifold learning approaches to learn reduced order dynamics. We observe that while manifold learning is generally capable of recovering reduced order dynamics, the quality of the learned representations varies substantially across different algorithms and hyperparameter choices. This is indicative of high sensitivity to the inherent geometric assumptions of the respective approaches and suggests a need for careful hyperparameter tuning, which can be expensive in practise. To overcome these challenges, we propose a framework for Automated Manifold Learning, which selects a manifold learning approach and corresponding hyperparameter choices based on representative subsamples of the input graph. We demonstrate that the proposed framework leads to performance gains both in scalability and in the learned representations' accuracy in capturing local and global geometric features of the underlying system dynamics.