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KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling
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Proposes KoopSTD, a Koopman-spectrum-based dynamical similarity metric that combines STFT timescale decoupling with residual-based (ResDMD-style) spectral mode filtering, and shows invariance to linear transformations and gains on Lorenz, attractor, RNN, fMRI, and LLM comparisons.
Problem
Measuring similarity between nonlinear dynamical systems that exhibit multi-timescale behavior is hard; existing Koopman-based and representation-based metrics are sensitive to cross-scale coupling and suffer from spectral pollution from spurious eigenvalues.
Approach
Apply STFT to decompose time series into time-frequency representations, perform DMD in the eigen-time-frequency coordinates from SVD, keep the top-r Koopman eigenvalues ranked by Galerkin/ResDMD-style squared relative residuals, and compare spectra with a permutation-invariant Wasserstein distance.
Key results
- On the Lorenz five-class task, KoopSTD achieves a Silhouette coefficient of 0.94 versus 0.47 for HAVOK-based DSA and negative values for CKA, Procrustes and cross-correlation.
- On noisy perceptual decision-making attractors, KoopSTD reaches a Silhouette of 0.99 versus 0.90 for HAVOK-based DSA.
- On the 3-bit Flip-Flop RNNs (where near-zero is preferred), KoopSTD scores -0.04 while Procrustes scores 0.98, indicating it recognizes the shared underlying dynamics.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use KoopSTD instead of HAVOK-based DSA | Lorenz systems · Lorenz systems | Silhouette coefficient: 0.47 -> 0.94 | Helped |
| Use KoopSTD instead of HAVOK-based DSA | PDM attractors · PDM attractors | Silhouette coefficient: 0.90 -> 0.99 | Helped |
| Use KoopSTD instead of CKA | 3-bit Flip-Flop RNNs · 3-bit Flip-Flop RNNs | Silhouette coefficient: 0.20 -> -0.04 | Helped |
| Add time-frequency transformation (timescale decoupling) (ablation) | KoopSTD · Lorenz systems | dissimilarity clustering | Helped |
| Add spectral residual control (ablation) | KoopSTD · Lorenz systems | dissimilarity clustering | Helped |
Surprising findings
- Procrustes analysis labels all attractor samples as identical, and CKA and cross-correlation perform worse than chance-level separation on Lorenz systems.
- Larger GPT-2 and LLaMA models show more concentrated, consistent representation dynamics, which the authors use to hypothesize a scaling-related consistency trend.
Limitations
- Computational complexity of the STFT plus residual computation may limit accessibility for large-scale use, per the impact statement.
- Spectral error estimation relies on Galerkin methods whose performance is constrained by basis expressiveness; alternative kernel-based error bounds are left to future work.
- Hyperparameters (window length, hop size, number of retained modes) are set per experiment, and the sensitivity analysis is limited.
Experimental setup
Datasets: Lorenz system simulations Perceptual decision-making attractors (Ostrow et al., 2024) 3-bit Flip-Flop RNN trajectories Narratives fMRI dataset (Nastase et al., 2021) Databricks Dolly 15K MMLU
Benchmarks: Lorenz system five-class clustering Perceptual decision-making attractors 3-bit Flip-Flop RNNs Auditory cortex fMRI (Narratives) GPT-2 and LLaMA scale comparison
Metrics: Silhouette coefficient intra- and inter-cluster distance
Baselines: CKA Procrustes analysis Cross-correlation HAVOK-based DSA
Builds on: Dynamical similarity analysis (HAVOK-based DSA) Residual dynamic mode decomposition (ResDMD) Koopman spectral kernel Generalized shape metrics
Closest papers by meaning
- Beyond DSA: Conjugacy-based Comparison of Dynamical Systems (NeurIPS 2026 · similarity 0.90)
- ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals (ICML 2025 · similarity 0.89)
- Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics (NeurIPS 2026 · similarity 0.88)
- Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions (NeurIPS 2025 · similarity 0.87)
- Spectral Analysis of Representational Similarity with Limited Neurons (NeurIPS 2025 · similarity 0.86)
- Identifying latent state transitions in non-linear dynamical systems (ICLR 2025 · similarity 0.86)
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