TopoPrimer: The Missing Topological Context in Forecasting Models

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.

Time-series foundation models can forecast across heterogeneous domains without task-specific training, but their forecasts are fixed once produced and cannot directly incorporate task-specific expert feedback. We study expert-guided forecast editing: a frozen foundation model generates candidate future trajectories, and an expensive expert evaluator scores them to guide forecast revision. Under a tight query budget, two natural strategies sit at opposite ends: best-of-$N$ purely exploits the foundation model's predictive distribution, while optimization approaches mostly explore the forecast horizon as an unstructured high-dimensional vector. Each extreme is individually sub-optimal. We introduce \textbf{DEFT}, an expert-guided forecast editing framework that balances the two by first exploiting the foundation model's predictive samples in a decomposed trend--seasonal space, then exploring around them via component-wise refinement. DEFT queries the expert only on complete trajectories, then reuses scores for the trend and seasonal components that appeared in the queried recombinations. This lets each expert query provide structured component-level feedback while keeping the foundation model frozen. We compare DEFT against direct search approaches, including best-of-$N$, cross-entropy methods, and Bayesian optimization, under matched expert-query budgets. Across two forecasting benchmarks consisting of 78 datasets, three time-series foundation models, four feedback types, and seven query budgets, DEFT consistently improves the effectiveness of expert guidance. A molecular-dynamics case study further suggests that the same principle extends to more physically grounded feedback, supporting the hypothesis that sparse test-time guidance should be spent balancing prior exploitation with structured exploration.