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Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data (2410.04814v2)

Published 7 Oct 2024 in cs.LG, cs.AI, math.DS, nlin.CD, and physics.data-an

Abstract: In science, we are often interested in obtaining a generative model of the underlying system dynamics from observed time series. While powerful methods for dynamical systems reconstruction (DSR) exist when data come from a single domain, how to best integrate data from multiple dynamical regimes and leverage it for generalization is still an open question. This becomes particularly important when individual time series are short, and group-level information may help to fill in for gaps in single-domain data. Here we introduce a hierarchical framework that enables to harvest group-level (multi-domain) information while retaining all single-domain characteristics, and showcase it on popular DSR benchmarks, as well as on neuroscience and medical data. In addition to faithful reconstruction of all individual dynamical regimes, our unsupervised methodology discovers common low-dimensional feature spaces in which datasets with similar dynamics cluster. The features spanning these spaces were further dynamically highly interpretable, surprisingly in often linear relation to control parameters that govern the dynamics of the underlying system. Finally, we illustrate transfer learning and generalization to new parameter regimes, paving the way toward DSR foundation models.

Citations (1)

Summary

  • The paper introduces a hierarchical model that separates domain-specific and group-level features to enhance dynamical system predictions.
  • It employs domain-specific recurrent neural networks trained with generalized teacher forcing, boosting stability in chaotic systems.
  • Experimental results on Lorenz benchmarks reveal superior state reconstruction and transfer learning while using fewer parameters than conventional methods.

Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data

The paper presents a hierarchical framework for learning interpretable models of dynamical systems from time series data, focusing on the integration of data from different dynamical regimes. This approach addresses a significant challenge in the field of dynamical systems reconstruction (DSR): how to effectively aggregate multi-domain information without losing domain-specific characteristics. The authors propose a method that efficiently combines group-level data while retaining individual system dynamics, facilitating transfer learning and generalization across diverse temporal domains.

Methodological Approach

The core innovation of the paper is a hierarchical model that separates domain-general and domain-specific features. By integrating subject-specific low-dimensional parameter vectors with group-level weights, the proposed model generates domain-specific recurrent neural networks (RNNs). These RNNs are trained end-to-end, utilizing generalized teacher forcing (GTF) methods which are crucial for handling chaotic systems and preventing exploding gradients. This hierarchical approach allows for the extraction of common feature spaces where similar dynamic datasets cluster, enabling both interpretability and prediction in unseen parameter regimes.

Experimental Results

The authors validate their approach on several benchmarks, including the Lorenz-63 and Lorenz-96 systems, demonstrating superior reconstruction of datasets with varying underlying dynamics. The method outperformed several state-of-the-art techniques, such as LEADS and CoDA, in both state space divergence and long-term temporal prediction accuracy. Notably, the hierarchical model required fewer parameters relative to these approaches, highlighting its efficiency and robustness.

Interpretability and Practical Implications

A key contribution of this work is its focus on interpretability. The hierarchy yields low-dimensional feature spaces that align with control parameters governing the dynamics of the systems under paper. For instance, in experiments with the Lorenz systems, the learned features showed linear correlations with bifurcation parameters, offering insights into the underlying dynamical structure.

The transfer learning capabilities demonstrate that the model can infer system dynamics from sparse data, crucial in fields like neuroscience and healthcare where data is costly to acquire. Additionally, the framework successfully discovers class-discriminative features in unsupervised settings. Experiments on EEG data illustrate the model's ability to outperform conventional time series feature extraction methods by leveraging dynamics-focused characteristics.

Theoretical and Future Directions

The hierarchical model proposed provides a promising avenue for research aimed at understanding dynamically complex systems across a variety of scientific domains. The approach challenges conventional methods by embedding dynamics into high-dimensional time series forecasting models, aligning with the broader objective of extracting interpretable models from complex data.

Moving forward, potential extensions include integrating Bayesian frameworks to quantify uncertainty in the parameters and exploring active learning paradigms for more efficient data use. The interpretability of learned features and their alignment with system bifurcations invites further research into the mathematical foundations of this characteristic. Exploring these avenues could enhance the application of hierarchical DSR models to even broader domains, potentially offering new insights into complex systems modeling.

By offering a robust, interpretable method for reconstructing and understanding dynamical systems from time series data, this paper lays foundational steps toward more generalizable and efficient learning frameworks in scientific computing.