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.

翻译：时间序列中的稀有事件对于建模至关重要，但由于数据稀疏而难以学习。现有生成模型在处理极端值时表现不佳。我们观察到，稀有事件会留下独特的拓扑指纹——即点云嵌入中贝蒂数的跃迁——这些特征比统计矩更稳定且更具判别性。我们提出PHINN，一种流匹配框架，使用动态贝蒂曲线作为条件信号，并采用持久性景观损失以确保同调一致性。该方法可扩展至多变量数据，包含一个用于设定贝蒂目标的自然语言接口，支持跨领域元学习和少样本生成，并提供了经过认证的对抗鲁棒性。在金融、流行病学和多模态基准测试中，PHINN在拓扑保真度（β-RMSE降低41-63%，跃迁准确率提升84%）上优于统计和扩散基线，且在尾部覆盖率上与跳扩散模型相当，同时在形状保真度上超越它们。所有结果均具有95%置信区间。