# TopoCast: A Topological Fidelity Framework for Evaluating Transformer-Based Time Series Forecasting

arXiv: https://arxiv.org/abs/2606.25439 · HTML: https://mlsanity.com/paper/2606.25439/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 24 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [Transformers in Time Series: A Survey](https://mlsanity.com/paper/2202.07125/) (2022)
- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Shape and Time Distortion Loss for Training Deep Time Series Forecasting Models](https://mlsanity.com/paper/1909.09020/) (2019)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Patch-wise Structural Loss for Time Series Forecasting](https://mlsanity.com/paper/2503.00877/) (2025)
- [TopoCL: Topological Contrastive Learning for Time Series](https://mlsanity.com/paper/2502.02924/) (2025)
- [Persistent Homology-induced Graph Ensembles for Time Series Regressions](https://mlsanity.com/paper/2503.14240/) (2025)
- [Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning](https://mlsanity.com/paper/2503.23757/) (2025)
- [Persistent homology of featured time series data and its applications](https://mlsanity.com/paper/2405.04796/) (2024)
- [Towards Measuring and Modeling Geometric Structures in Time Series Forecasting via Image Modality](https://mlsanity.com/paper/2507.23253/) (2025)
- [Are We Winning the Wrong Game? Revisiting Evaluation Practices for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2603.08156/) (2026)

## Abstract (arXiv)

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.