# PHINN: Persistent Homology Inspired Neural Network for Rare-Event Time Series Generation

arXiv: https://arxiv.org/abs/2606.15452 · HTML: https://mlsanity.com/paper/2606.15452/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 5 of its 29 references that are in the map.
- First posted on arXiv 13 Jun 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 5 of 5)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [Kernel Embeddings and the Separation of Measure Phenomenon](https://mlsanity.com/paper/2505.04613/) (2025)

## Abstract (arXiv)

Rare events in time series are critical to model but hard to learn due to data scarcity. Current generative models struggle with extreme values. We observe that rare events leave distinct topological fingerprints - transitions in Betti numbers from point-cloud embeddings - that are more stable and discriminative than statistical moments. We introduce PHINN, a flow-matching framework using dynamic Betti curves as conditioning signals and a persistence landscape loss for homology consistency. It scales to multivariate data, includes a natural-language interface to set Betti targets, supports cross-domain meta-learning and few-shot generation, and provides certified adversarial robustness. On financial, epidemiological, and multi-modal benchmarks, PHINN outperforms statistical and diffusion baselines in topological fidelity (beta-RMSE down 41-63%, transition accuracy up 84%) and matches jump-diffusion models in tail coverage while exceeding them in shape fidelity. All results have 95% confidence intervals.