TopTimeNet: Topologically-assisted time-series classification model

Sharareh Sayyad; Sophia Bazzi

cs.LGSep 30, 2026

TopTimeNet: Topologically-assisted time-series classification model

Authors: Sharareh Sayyad, Sophia Bazzi

Organizations: Department of Mathematics and Statistics, Washington State University, Pullman, Washington 99164-3113, USA · European Molecular Biology Laboratory, EMBL Hamburg, c/o DESY, Building 25A, Notkestraße 85, 22603 Hamburg, Germany

Abstract

Distinguishing periodic from chaotic dynamics in a time series is a fundamental challenge in both physics and engineering. Yet, end-to-end learned architectures must discover both a representation and a decision boundary from data, at substantial cost. We introduce TopTimeNet, which decouples these tasks: a fixed, non-learned stage extracts a $42$ -dimensional geometric and topological descriptor from Takens delay embeddings and persistent homology, and a lightweight learnable stage performs classification. On a benchmark of $49$ nonlinear dynamical systems, a $1{,}638$ -parameter configuration matches the mean accuracy of one with $33\times$ more trainable parameters. Additionally, this approach delivers mean accuracy comparable to convolutional neural networks and surpasses the average performance of converged Transformer models, while requiring three to four orders of magnitude fewer trainable parameters. Robustness also depends sharply on where noise is introduced: TopTimeNet degrades gracefully under perturbations to its precomputed features, but degrades sharply when noise is introduced into the raw signal and the full feature-extraction pipeline is recomputed, showing that robustness to perturbations of the precomputed features does not imply robustness of the complete raw-signal-to-prediction pipeline. These results show that decoupling fixed geometric and topological feature construction from a lightweight discriminative stage can achieve comparable classification accuracy with substantially fewer trainable parameters.

Figures & tables

Figure 1: Schematic illustration of the Top Time Net algorithm. The segmented time-series data first pass through a non-trainable feature-extraction stage of the algorithm, where point clouds are generated and geometric and topological statistics are extracted. This process creates a complete feature set that will be used in subsequent steps. The five groups of extracted features are then processed through a Group Projector and combined using a Fusion Layer. Finally, the combined features are classified using a Classifier Head to produce the final class logits.

Figure 2: Time-dependent

\theta_{1}

and its Takens embedding. Left panels present

\theta_{1}(t)

for the periodic (top) and chaotic (bottom) regimes. Right panels show the associated point clouds obtained via a Takens embedding with time delay

\tau=20

and embedding dimension

d_{\rm emb}=2

Feature	Periodic	Chaotic
Diameter	$1.1552$	$3.2792$
Correlation-dimension proxy	$1.0672$	$1.6585$
Mean nearest-neighbor dist.	$0.0014$	$0.0410$
Std. nearest-neighbor dist.	$0.0005$	$0.0293$

Table 1: Geometric branch feature values for the double pendulum, periodic vs. chaotic regime.

Figure 3: Persistence diagrams for the two tracked homology dimensions.

H_{0}

(connected components, circles) and

H_{1}

(loops, triangles) features are shown for the periodic (left) and chaotic (right) point clouds of Fig. 2 ; note the different axis ranges between panels. Points farther from the diagonal correspond to more persistent topological features.

Feature	Periodic	Chaotic
Entropy, $H_{0}$	$5.5456$	$6.6991$
Entropy, $H_{1}$	$0.0379$	$5.0058$

Table 2: Entropy branch feature values for the double pendulum, periodic vs. chaotic regime.

	$H_{0}$		$H_{1}$
Feature	Periodic	Chaotic	Periodic	Chaotic
Maximum lifetime	$0.9532$	$0.4576$	$0.9348$	$0.1533$
Total lifetime	$4.8949$	$52.1417$	$0.9386$	$6.0242$
Dominance ratio	$0.1947$	$0.0088$	$0.9959$	$0.0254$
Coefficient of variation	$6.1556$	$0.6769$	$5.0775$	$1.0785$
Number of significant lifetimes	$1$	$836$	$1$	$169$

Table 3: Lifetime branch feature values for the double pendulum, periodic vs. chaotic regime.

Figure 4: Diameter-normalized Betti curves

\beta(\varepsilon)

for

H_{0}

(top) and

H_{1}

(bottom), periodic (left) and chaotic (right). Note the different vertical scales between panels.

Figure 5: Persistence images for $H_{0}$ (top) and $H_{1}$ (bottom), periodic (left) and chaotic (right). Color indicates pixel intensity (note the different color scales between panels); axes are birth and persistence pixel bins.

	$H_{0}$		$H_{1}$
Feature	Periodic	Chaotic	Periodic	Chaotic
Max	$980.0000$	$980.0000$	$1.0000$	$49.0000$
Mean	$20.4000$	$25.3800$	$0.8000$	$1.6400$
Standard deviation	$137.0863$	$140.7353$	$0.4000$	$7.1882$
Peak position	$0.0000$	$0.0000$	$0.0204$	$0.0408$
Turning points	$0$	$0$	$0$	$1$

Table 4: Betti curve branch feature values for the double pendulum, periodic vs. chaotic regime.

	$H_{0}$		$H_{1}$
Feature	Periodic	Chaotic	Periodic	Chaotic
Total mass	$1.2801$	$34.7081$	$0.2390$	$5.0567$
Max pixel	$0.0125$	$0.1561$	$0.0030$	$0.0173$
Active fraction	$0.2500$	$0.5000$	$0.3000$	$0.6000$
Birth centroid	$0.5000$	$0.5000$	$0.5000$	$0.4265$
Persistence centroid	$0.2542$	$0.3124$	$0.8928$	$0.4860$

Table 5: Persistence image branch feature values for the double pendulum, periodic vs. chaotic regime.

	Small model	Large model
Trainable parameters	$1{,}638$	$54{,}886$
Embedding dim. $D$	$16$	$128$
Fusion strategy	Bilinear	Bilinear
Classifier head	$()$	$(64,32,16)$
Activation	GELU	Leaky ReLU
Dropout	$0.05$	$0.0$

Table 6: Comparison of the small and large Top Time Net configurations selected by the hyperparameter search of Sec. II.3.2 . All test metrics are mean

\pm

std over

30

independent training runs. Both configurations use bilinear fusion (Sec. II.2.2 ), which has no rank or attention hyperparameters; the rank/attention-layer/attention-head values sampled elsewhere in the search do not apply to either selected configuration.

	Top Time Net (small)	CNN ( $n=10$ )	Transformer ( $n=7$ )
Trainable parameters	$1{,}638$	$1{,}824{,}898$	$33{,}435{,}570$
Accuracy	$0.9758\pm 0.0029$	$0.9708\pm 0.0110$	$0.9412\pm 0.0133$
F1 score	$0.9758\pm 0.0029$	$0.9708\pm 0.0110$	$0.9412\pm 0.0133$
G-mean	$0.9757\pm 0.0029$	$0.9708\pm 0.0111$	$0.9411\pm 0.0133$
Precision	$0.9758\pm 0.0029$	$0.9708\pm 0.0110$	$0.9412\pm 0.0133$
Recall	$0.9758\pm 0.0029$	$0.9710\pm 0.0109$	$0.9416\pm 0.0132$

Table 7: Top Time Net (small configuration) versus CNN and Transformer baselines. Top Time Net and CNN report mean

\pm

std over

30

and

10

independent training runs, respectively, with no failed or collapsed runs in either case. For the Transformer,

10

independent training attempts were performed; the reported mean

\pm

std values are computed over the

7

runs that converged, while the remaining

3

attempts remained at a near-chance validation-accuracy plateau and are reported as non-converged (see text).

Figure 6: Noise robustness of all three models. (a) Test accuracy and (b) Expected Calibration Error (ECE,

M=10

equal-width bins), both as a function of the Gaussian noise standard deviation

\sigma

. For Top Time Net, noise is injected at two distinct points: directly into the precomputed

42

-dimensional feature vector of Sec. II.2.1 (“feature-level”), and into the raw time series itself, with the entire feature-extraction pipeline, Takens embedding, persistent homology, and all five feature branches, recomputed from the corrupted signal (“raw-signal”). The CNN and Transformer baselines have no intermediate feature representation to perturb separately, so only their raw-signal curves are shown. Thick lines show the mean over independent training runs (

n=30

for Top Time Net,

n=10

for the CNN,

n=7

for the Transformer’s converged runs), shaded bands show

\pm 1

standard deviation, and thin lines show individual runs.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Raw double-pendulum time series under increasing Gaussian noise. Top: periodic. Bottom: chaotic.

Figure 8: Takens-embedded point clouds under raw-signal Gaussian noise. Top: periodic. Bottom: chaotic.

Figure 9: Persistence diagrams under raw-signal Gaussian noise. Top: periodic. Bottom: chaotic.

Figure 10: Relative feature-vector drift

\|f_{\sigma}-f_{0}\|_{2}/\|f_{0}\|_{2}

under raw-signal Gaussian noise , for the periodic and chaotic double-pendulum examples of Figs. 7 – 9 .

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Open paper AskPDF arXiv

Scientific time series often encode predictive geometric structure, including connectivity, cycles, shell-like geometry, directional changes, and nonlinear neighborhoods, that standard dot-product attention does not explicitly represent. We introduce a topology-aware attention framework that adds such structure to attention logits using persistent homology (H0-H2), anchored Euler characteristic transforms, and kernel-Hilbert channels. A validation-gated local residual captures local topological signals, including a Zeng-style local H0 component, only when held-out validation data support the correction. Exact Vietoris-Rips computations and smooth topological surrogates are evaluated under a no-leakage protocol with train-only calibration, validation-only selection, and test-only reporting. We evaluate guarded topology-aware variants across three architecture families: lightweight attention/Ridge, PatchTSTForRegression, and TimeSeriesTransformerForPrediction. Experiments include synthetic benchmarks isolating higher-order topology and real datasets covering CO2, S&P 500 return-window geometry, and NASA IMS bearing degradation. The audit uses matched paired comparisons across seven dataset units, three random seeds, and three chronological splits, giving 63 paired units per architecture and 189 paired units overall. Topology-aware models show positive paired effects when geometry is predictive, with heterogeneous magnitude across datasets and architectures. Lightweight attention/Ridge improves in 46 of 63 units, with mean relative RMSE reduction of 12.5% and paired randomization p=7.2e-4; PatchTST improves in 33 units and retains the baseline in 20 units, with 23.5% reduction and p=3.5e-5; and TimeSeriesTransformer improves in 47 units, with 47.8% reduction and p<1e-4. The results support topology as a validation-selected, architecture-compatible inductive bias.

Abstract

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