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Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data
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Introduces a hierarchical framework that maps low-dimensional subject-specific feature vectors to shared group-level parameters of a piecewise-linear RNN (shPLRNN) for dynamical systems reconstruction across multiple dynamical regimes, trained with generalized teacher forcing.
Problem
Generative dynamical systems reconstruction from multiple short time series spanning different dynamical regimes, where individual series are too short to train good models alone, and how to integrate multi-domain data for generalization.
Approach
Subject-specific latent feature vectors are linearly projected through shared group-level matrices to produce all parameters of a shPLRNN for each subject, trained end-to-end with generalized teacher forcing; the learned feature spaces turn out to be interpretable and often linearly related to control parameters.
Key results
- On Lorenz-63 (64 subjects), hier-shPLRNN reaches D_stsp 0.394±0.014 and D_H 0.097±0.005, versus 1.82±0.16 and 0.118±0.002 for the shPLRNN ensemble, and 1.4±0.4 for CoDA.
- Unsupervised GMM clustering in the learned DS feature space recovers epileptic vs healthy EEG labels with 92.6±1.2% accuracy, outperforming tsfresh, Catch-22, ROCKET, MiniRocket and autoencoders.
- Learned single feature for Lorenz-63 and Rössler is highly predictive of the control parameters (rho, c) with a clearly linear relation; feature vectors predict pulse-wave haemodynamic parameters with mean linear R^2=0.83.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use hierarchical rather than ensemble shPLRNNs (ablation) | shPLRNN · Lorenz-63 | D_stsp: 1.82±0.16 -> 0.394±0.014 | Helped |
| use hierarchical rather than ensemble shPLRNNs (ablation) | shPLRNN · Lorenz-96 | D_stsp: 1.291±0.023 -> 0.580±0.002 | Helped |
| replace shPLRNN with Vanilla RNN (ablation) | hierarchical DSR · Lorenz-63 | D_stsp: 0.394±0.014 -> 0.57±0.09 | Mixed |
| replace shPLRNN with LSTM (ablation) | hierarchical DSR · Lorenz-63 | D_stsp: 0.394±0.014 -> 2.3±0.3 | Hurt |
| add shared linear observation model (ablation) | hier-shPLRNN · Lorenz-63 | D_H: 0.097±0.005 -> 0.087±0.012 | No clear effect |
| add feature-conditioned linear observation model (ablation) | hier-shPLRNN · Lorenz-63 | D_H: 0.097±0.005 -> 0.087±0.010 | No clear effect |
| replace naive linear mapping with outer product parameterization (ablation) | hier-shPLRNN · Lorenz-63 | D_stsp: 0.394±0.014 -> 0.36±0.03 | No clear effect |
| replace naive linear mapping with MLP mapping (ablation) | hier-shPLRNN · Lorenz-63 | D_stsp: 0.394±0.014 -> 0.85±0.09 | Hurt |
Surprising findings
- Training time to reach a performance criterion decreases as the number of subjects increases.
- Learned feature spaces are often linearly related to ground-truth control parameters even though the model is piecewise-linear and the ground truth is polynomial.
Limitations
- Mechanistic basis for why the learned features align with control parameters is unclear.
- Currently limited to dynamical systems evolving on similar time scales.
- Linear group-level mapping works best; more expressive MLP mapping performed clearly worse.
Experimental setup
Datasets: Lorenz-63 Rössler Lorenz-96 Chua's circuit arterial pulse wave simulations clinical EEG (epileptic vs healthy) fMRI time series electricity consumption
Benchmarks: Lorenz-63 Lorenz-96 Rössler EEG classification arterial pulse wave
Metrics: state space divergence D_stsp Hellinger distance on power spectra D_H clustering accuracy R^2
Baselines: shPLRNN ensemble CoDA LEADS tsfresh Catch-22 ROCKET MiniRocket attention-based convolutional autoencoder
Builds on: shallow PLRNN generalized teacher forcing LEADS CoDA
Closest papers by meaning
- Out-of-Domain Generalization in Dynamical Systems Reconstruction (ICML 2024 · similarity 0.92)
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- A Minimal Interpretable Architecture for Zero-Shot Reconstruction of Dynamical Systems (NeurIPS 2026 · similarity 0.89)
- True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics (NeurIPS 2025 · similarity 0.89)
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