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.

Modeling hierarchical latent dynamics behind time series data is critical for capturing temporal dependencies across multiple levels of abstraction in real-world tasks. However, existing temporal causal representation learning methods fail to capture such dynamics, as they fail to recover the joint distribution of hierarchical latent variables from single-timestep observed variables. Interestingly, we find that the joint distribution of hierarchical latent variables can be uniquely determined using three conditionally independent observations. Building on this insight, we propose a Causally Hierarchical Latent Dynamic (CHiLD) identification framework. Our approach first employs temporal contextual observed variables to identify the joint distribution of multi-layer latent variables. Sequentially, we exploit the natural sparsity of the hierarchical structure among latent variables to identify latent variables within each layer. Guided by the theoretical results, we develop a time series generative model grounded in variational inference. This model incorporates a contextual encoder to reconstruct multi-layer latent variables and normalize flowbased hierarchical prior networks to impose the independent noise condition of hierarchical latent dynamics. Empirical evaluations on both synthetic and realworld datasets validate our theoretical claims and demonstrate the effectiveness of CHiLD in modeling hierarchical latent dynamics.

Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak topology may cause huge loss in multi-periods decision making problems. To address such discontinuities, Aldous introduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however, it was considered to be hard to find efficient numerical implementations. In this paper, we introduce a novel metric called High Rank PCF Distance (HRPCFD) for extended weak convergence based on the high rank path development method from rough path theory, which also defines the characteristic function for measure-valued processes. We then show that such HRPCFD admits many favourable analytic properties which allows us to design an efficient algorithm for training HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Our numerical experiments on both hypothesis testing and generative modelling validate the out-performance of our approach compared with several state-of-the-art methods, highlighting its potential in broad applications of synthetic time series generation and in addressing classic financial and economic challenges, such as optimal stopping or utility maximisation problems.

To address this shortcoming of weak convergence, D. Aldous [

Time series data plays a pivotal role in a wide variety of fields but faces challenges related to privacy concerns. Recently, synthesizing data via diffusion models is viewed as a promising solution. However, existing methods still struggle to capture long-range temporal dependencies and complex channel interrelations. In this research, we aim to utilize the sequence modeling capability of a State Space Model called Mamba to extend its applicability to time series data generation. We firstly analyze the core limitations in State Space Model, namely the lack of consideration for correlated temporal lag and channel permutation. Building upon the insight, we propose Lag Fusion Mamba and Permutation Scanning Mamba, which enhance the model's ability to discern significant patterns during the denoising process. Theoretical analysis reveals that both variants exhibit a unified matrix multiplication framework with the original Mamba, offering a deeper understanding of our method. Finally, we integrate two variants and introduce Diffusion Mamba for Time Series (DiM-TS), a high-quality time series generation model that better preserves the temporal periodicity and inter-channel correlations. Comprehensive experiments on public datasets demonstrate the superiority of DiM-TS in generating realistic time series while preserving diverse properties of data.

Time series are ubiquitous in many applications that involve forecasting, classification and causal inference tasks, such as healthcare, finance, audio signal processing and climate sciences. Still, large, high-quality time series datasets remain scarce. Synthetic generation can address this limitation; however, current models confined either to the time or frequency domains struggle to reproduce the inherently multi-scaled structure of real-world time series. We introduce WaveletDiff, a novel framework that trains diffusion models directly on wavelet coefficients to exploit the inherent multi-resolution structure of time series data. The model combines dedicated transformers for each decomposition level with cross-level attention mechanisms that enable selective information exchange between temporal and frequency scales through adaptive gating. It also incorporates energy preservation constraints for individual levels based on Parseval's theorem to preserve spectral fidelity throughout the diffusion process. Comprehensive tests across six real-world datasets from energy, finance, and neuroscience domains demonstrate that WaveletDiff consistently outperforms state-of-the-art time-domain and frequency-domain generative methods on both short and long time series across five diverse performance metrics. For example, WaveletDiff achieves discriminative scores and Context-FID scores that are $3\times$ smaller on average than the second-best baseline across all datasets.

In industrial equipment monitoring, fault diagnosis is critical for ensuring system reliability and enabling predictive maintenance. However, the scarcity of fault data, due to the rarity of fault events and the high cost of data annotation, significantly hinders data-driven approaches. Existing time-series generation models, optimized for abundant normal data, struggle to capture fault distributions in few-shot scenarios, producing samples that lack authenticity and diversity due to the large domain gap and high intra-class variability of faults. To address this, we propose a novel few-shot fault time-series generation framework based on diffusion models. Our approach employs a positive-negative difference adapter, leveraging pre-trained normal data distributions to model the discrepancies between normal and fault domains for accurate fault synthesis. Additionally, a diversity loss is introduced to prevent mode collapse, encouraging the generation of diverse fault samples through inter-sample difference regularization. Experimental results demonstrate that our model significantly outperforms traditional methods in authenticity and diversity, achieving state-of-the-art performance on key benchmarks.