Time-series forecasting via topological information supervised framework with efficient topological feature learning

Topological Data Analysis (TDA) has emerged as a powerful tool for extracting meaningful features from complex data structures, driving advancements in neuroscience, biology, machine learning, and financial modeling. However, its integration with time-series forecasting remains underexplored due to three key challenges: limited use of temporal dependencies in topological features, computational bottlenecks in persistent homology, and the deterministic nature of TDA pipelines that restrict generalized learning. To address these issues, we propose the Topological Information Supervised (TIS) Prediction framework, which employs neural networks and Conditional Generative Adversarial Networks (CGANs) to generate synthetic topological features that preserve distributional properties while reducing computation time. A novel training strategy incorporating a topological consistency loss further enhances predictive accuracy. We evaluate TIS across recurrent and Transformer-based models, showing that integrating topological information yields consistent performance improvements across diverse architectures and forecasting horizons. While the magnitude of gains varies, tending to be more modest for recent architectures, the results highlight TIS as a lightweight, complementary inductive bias that augments existing models beyond architectural refinements. This work advances TDA-based time-series prediction and opens new directions for embedding topological insights into deep learning frameworks.