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Beyond DSA: Conjugacy-based Comparison of Dynamical Systems

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: State-space inference for stochastic systems · 0 citations (OpenAlex)

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Shows that orthogonal alignment of Koopman matrices (DSA) is neither necessary nor sufficient for topological conjugacy, and proposes Conjugacy-based Similarity Analysis (CSA), which restricts alignments to projected composition operators induced by candidate bijections.

Problem

Dynamical Similarity Analysis compares Koopman approximations via orthogonal alignment, which may not correspond to a valid state-space relabelling, so its distances do not reliably indicate topological conjugacy between dynamical systems.

Approach

Replace the orthogonal alignment search of DSA with a search over candidate state-space bijections h_theta; for each, fit a finite-data composition-operator matrix P_theta from observable dictionaries and score the intertwining residual ||P F - G P|| / ||P||.

Key results

• In the state-space conjugacy experiment, DSA's Pearson correlation falls below 0.5 across theta in [0,0.30] while CSA's correlation stays close to 1.
• Cross-dictionary Fourier vs Legendre comparison: CSA nRMSE 0.212 and correlation 0.977 versus DSA nRMSE 0.741 and correlation 0.671.
• Hankel-DMD experiment: CSA-Krylov distance stays under about 0.08 across sweeps while DSA-Krylov rises to 0.5-0.8.

Surprising findings

• Orthogonal (DSA) alignment can make genuinely non-conjugate systems look similar, e.g., Bernoulli shifts with different alphabet sizes are unitarily equivalent Koopman operators yet not conjugate.
• Typical smooth conjugacies such as x + theta sin(pi x) induce non-unitary composition operators, so DSA's orthogonal class cannot capture them.

Limitations

• CSA is not a formal metric and its conclusions depend on the supplied candidate bijection family and observable spaces.
• Experiments are controlled: the true bijection family is given to CSA, so misspecification is not tested.
• Experiments are restricted to one-dimensional logistic and linear maps with EDMD and Hankel-DMD.

Experimental setup

Datasets: synthetic logistic map trajectories synthetic contracting linear map trajectories

Benchmarks: controlled logistic-map conjugacy sweeps Hankel-DMD/Krylov conjugacy sweeps

Metrics: normalized affine RMSE Pearson correlation distance flatness under conjugacy

Baselines: DSA DSA-Krylov

Builds on: Dynamical Similarity Analysis EDMD Hankel-DMD

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