PHINN Generates Rare Events in Time Series Using Topological Fingerprints

Modeling rare events in time series data presents a significant challenge for generative models due to the inherent scarcity of such events. Existing methods often struggle to accurately capture extreme values, which are critical for understanding and predicting high-impact occurrences across various domains. Researchers have introduced PHINN (Persistent Homology Inspired Neural Network), a novel flow-matching framework designed to address this limitation. PHINN capitalizes on the observation that rare events leave distinct topological fingerprints, specifically transitions in Betti numbers derived from point-cloud embeddings. These topological signals are more stable and discriminative than traditional statistical moments, providing a robust conditioning signal for the generative process. PHINN incorporates dynamic Betti curves as conditioning signals and utilizes a persistence landscape loss to ensure homology consistency. The framework is scalable to multivariate data, features a natural language interface for setting Betti targets, supports cross-domain meta-learning, and offers certified adversarial robustness. Empirical evaluations on financial, epidemiological, and multi-modal datasets demonstrate that PHINN significantly outperforms statistical and diffusion baselines in topological fidelity and matches jump-diffusion models in tail coverage while exceeding them in shape fidelity.