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Let's do the time-warp-attend: Learning topological invariants of dynamical systems
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Proposes a topologically-augmented, self-attention convolutional classifier trained on a single simple oscillator to classify pre- vs post-Hopf dynamical regimes, generalizing to unseen systems, bifurcation boundaries, a repressilator, and pancreas single-cell RNA velocity data.
Problem
Classifying qualitative dynamical regimes (fixed point vs limit cycle) and locating bifurcation boundaries across diverse dynamical systems and real noisy data, where governing equations are unknown and time-series methods do not generalize.
Approach
Train a SAGAN-style convolutional attention classifier on angular representations of vector fields from a simple oscillator, augmented with random monotonic rational-quadratic spline diffeomorphisms that preserve dynamical class, so the learned features are invariant to geometric warping but selective for topological structure.
Key results
- Average test accuracy under sigma=0.1 Gaussian noise of 0.87 across eight systems, versus 0.80 for the Autoencoder baseline and 0.72 for Lyapunov.
- Achieves 87% accuracy classifying repressilator vector fields and 94% accuracy predicting cyclic behavior matching high cell-cycle scores in pancreas single-cell data, versus 54% for Critical Points.
- Accuracy on Augmented SO degrades gracefully with noise (0.54 at sigma=1.0) whereas Critical Points drops to 0.59 at sigma=0.1.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add topological data augmentations (ablation) | Our Model · Complex synthetic systems | accuracy | Helped |
| use angular representation instead of vector representation (ablation) | Our Model · Complex synthetic systems | accuracy | Helped |
| add self-attention layers (ablation) | Our Model · Complex synthetic systems | accuracy | Helped |
| remove augmentations, use vector representation, and remove attention (ablation) | Our Model · Complex synthetic systems | accuracy | Hurt |
| increase Gaussian noise on Augmented SO from σ=0 to σ=1.00 | Our Model · Augmented SO | accuracy: 0.93±0.01 -> 0.54±0.02 | Hurt |
| train Our Model with learning rate 1e-5 instead of the default (ablation) | Our Model · Complex synthetic systems, average | accuracy: 0.87 -> 0.79 | Hurt |
| train Our Model for 40 epochs instead of the default (ablation) | Our Model · Complex synthetic systems, average | accuracy: 0.87 -> 0.88 | Helped |
Surprising findings
- Critical Points achieves near-perfect accuracy on noiseless data but drops to 56% on Supercritical Hopf with sigma=0.1 Gaussian noise.
- Selkov upper bifurcation boundary is undetected because damping produces slow spirals indistinguishable from limit cycles numerically.
Limitations
- Confined to two-dimensional dynamics; higher dimensions require dimensionality reduction and face scaling challenges.
- Supervised multi-class setup; performance on Selkov and BZ systems is weaker, with Selkov accuracy at 0.65.
- Misclassifies damped systems near bifurcation boundaries as cyclic (Selkov upper boundary).
Experimental setup
Datasets: Simple oscillator augmented with rational-quadratic spline diffeomorphisms (Augmented SO) pancreatic endocrinogenesis scRNA-seq data (~3,600 cells) repressilator simulations
Benchmarks: SO Augmented SO Supercritical Hopf Lienard Polynomial Lienard Sigmoid Van der Pol BZ Reaction Selkov repressilator pancreas scRNA-seq
Metrics: classification accuracy mean and standard deviation over 50 runs
Baselines: Critical Points Lyapunov Phase2vec Autoencoder Parameters
Builds on: SAGAN rational-quadratic neural spline flows phase2vec
Closest papers by meaning
- TRENDy: Temporal Regression of Effective Nonlinear Dynamics (ICLR 2025 · similarity 0.89)
- Context-Informed Neural ODEs Unexpectedly Identify Broken Symmetries: Insights from the Poincaré–Hopf Theorem (ICML 2025 · similarity 0.87)
- Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction (NeurIPS 2024 · similarity 0.86)
- Learning the Topology and Behavior of Discrete Dynamical Systems (AAAI 2024 · similarity 0.86)
- Benchmarking Structural Inference Methods for Interacting Dynamical Systems with Synthetic Data (NeurIPS 2024 · similarity 0.86)
- Identifying latent state transitions in non-linear dynamical systems (ICLR 2025 · similarity 0.86)
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