Organizations: Department of Computer Science and Engineering, University of Moratuwa, Katubedda, Moratuwa, 10400, Sri Lanka.
Abstract
Deep learning-based models have achieved state-of-the-art performance in Time Series Forecasting (TSF), yet their evaluation remains dominated by pointwise error metrics such as Mean Squared Error (MSE), which quantify numerical accuracy but overlook structural properties of the forecast signal, including recurrent dynamics, oscillatory behavior, and phase alignment. As a result, forecasts exhibiting over-smoothing, phase shifts, or frequency distortions may achieve favorable error scores despite substantial structural degradation. To address this limitation, we propose TopoCast, a topology-driven framework for evaluating structural fidelity in TSF. TopoCast reconstructs phase-space representations of forecast and ground-truth sequences using Takens delay embedding and applies persistent homology to characterize their intrinsic dynamics. We derive four complementary topological fidelity measures from persistence diagrams and aggregate them into a Topological Fidelity Score (TFS). We further introduce dominant cycle overlap, a novel metric that maps persistent topological features to the temporal domain to assess whether dominant oscillatory patterns occur at the correct time points. Combined with TFS, this yields the Localized Topological Fidelity Score (LTFS), a phase-aware measure that captures temporal localization errors invisible to existing evaluation metrics. Experiments on five Transformer architectures across three real-world benchmark datasets demonstrate that models with similar forecasting errors can exhibit markedly different structural fidelity profiles, revealing failure modes overlooked by conventional evaluation and highlighting the value of topology-aware forecast assessment.
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.
We introduce TopoPrimer, a framework that makes the global topological structure of the series population an explicit input to any forecasting model. TopoPrimer improves accuracy across diverse domains, stabilizes forecasts under seasonal demand spikes, and closes the cold-start gap. Precomputed once per domain via persistent homology and spectral sheaf coordinates, TopoPrimer deploys per token for fully-trained models and as a lightweight adapter for pre-trained backbones. Of these two components, sheaf coordinates are the primary accuracy driver. Across four public benchmarks on Chronos and TimesFM, TopoPrimer consistently improves forecasting accuracy, with gains of up to 7.3% MSE on ECL. The topology advantage persists with near-identical magnitude across zero-shot and fine-tuned backbones, suggesting topology and per-series training capture complementary signals. The gains are most pronounced in difficult regimes. Under peak seasonal demand, classical and zero-shot models degrade by up to 50%, while TopoPrimer stays within 10%. At cold start with no item history, TopoPrimer reduces MAE by 27% over a topology-free baseline.
Time series forecasting (TSF) underpins consequential decisions in energy, transportation, finance, and healthcare, yet TSF models are almost universally ranked by a single number (e.g., average error) on clean held-out data, under the implicit assumption that it predicts deployed reliability. However, real faults are not i.i.d noise but structured events with temporal shape, broken cross-variable dependencies, regime change coupled with missingness, and causal propagation across a sensing pipeline. Treating TSF robustness as a data-quality problem, we present TS-Fault, a benchmark that evaluates forecasting models under explicit, parameterized fault scenarios with controllable semantic difficulty. TS-Fault organizes recurring failures into four modes along two orthogonal axes (observation- vs mechanism-level; univariate vs multivariate) and injects each fault into the most prediction-critical window via a unified importance score. This design enables robustness to be tested against the structures models actually rely on, rather than reduced to generic noise sensitivity. We evaluate 21 models across 6 datasets, 4 modes, and 5 difficulty levels under a paired clean/corrupt protocol. The results reveal three findings that contradict common leaderboard intuition: (i) clean-data accuracy anti-correlates with robustness; (ii) clean rankings are preserved under observation-level faults but reshuffled under mechanism-level faults; and (iii) all catastrophic failures occur under mechanism-level faults, with foundation models achieving the highest clean-data accuracy yet exhibiting the greatest fragility. The code is publicly available at https://github.com/Ray-zyy/TS-Fault.