Latent Diffeomorphic Dynamic Mode Decomposition

2025-05-09

AI Summary

To address the challenge of jointly achieving interpretability and long-term prediction accuracy in nonlinear dynamical system modeling, this paper proposes a latent-space diffeomorphic dimensionality reduction method integrating Dynamic Mode Decomposition (DMD) with Gated Recurrent Units (GRUs). The key innovation lies in the first incorporation of diffeomorphic constraints into the DMD framework, enabling explicit modeling of strong nonlinearities and long-range temporal dependencies while preserving linearly interpretable latent dynamics. By jointly optimizing latent-space encoding, diffeomorphic manifold learning, and recurrent memory mechanisms, the method achieves significant improvements in multi-step streamflow forecasting across watersheds. Experimental results demonstrate its effectiveness, generalizability, and physical interpretability on real-world nonlinear time-series systems, bridging the gap between data-driven prediction and mechanistic understanding.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Natural Language Processing: Language Grounding & Multi-modal NLP
• Data Mining & Knowledge Management: Data Stream Mining

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for 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

We present Latent Diffeomorphic Dynamic Mode Decomposition (LDDMD), a new data reduction approach for the analysis of non-linear systems that combines the interpretability of Dynamic Mode Decomposition (DMD) with the predictive power of Recurrent Neural Networks (RNNs). Notably, LDDMD maintains simplicity, which enhances interpretability, while effectively modeling and learning complex non-linear systems with memory, enabling accurate predictions. This is exemplified by its successful application in streamflow prediction.

Problem

• Analyzing non-linear systems with interpretability and predictive power
• Modeling complex non-linear systems with memory for accurate predictions
• Applying data reduction for streamflow prediction

Innovation

• Combines DMD interpretability with RNN predictive power
• Models complex non-linear systems with memory
• Enhances interpretability while maintaining simplicity