Automated Manifold Learning for Reduced Order Modeling

2025-06-02

AI Summary

In spatiotemporal data-driven system dynamics modeling, manifold learning algorithms suffer from heavy reliance on manual hyperparameter tuning and insufficient robustness in geometric representation. To address these issues, this paper proposes an automated manifold learning framework. First, it constructs a spatiotemporal proximity graph; then, it integrates classical methods (t-SNE, UMAP) with graph neural networks and introduces a meta-learning-driven subgraph sampling strategy to enable adaptive algorithm selection and joint hyperparameter optimization. Compared to conventional approaches, the framework significantly reduces human intervention while enhancing the fidelity of low-dimensional manifolds in capturing both local and global geometric structures of dynamical systems. Experiments across diverse dynamical systems demonstrate that the learned low-dimensional representations achieve superior accuracy and scalability, effectively supporting subsequent model order reduction and dynamic discovery.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Identifying geometric structure in spatial-temporal data
• Learning reduced order dynamics via manifold learning
• Automating algorithm and hyperparameter selection for better performance

Innovation

• Automated Manifold Learning framework for dynamics
• Spatial-temporal proximity graph encodes similarity
• Hyperparameter tuning via representative subsamples