Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.
We propose a novel framework for learning time-varying graphs from spatiotemporal measurements. Given an appropriate prior on the temporal behavior of signals, our proposed method can estimate time-varying graphs from a small number of available measurements. To achieve this, we introduce three regularization terms in convex optimization problems that constrain the sparseness of temporal variations of the time-varying networks. Moreover, a computationally scalable algorithm is introduced to solve the optimization problem efficiently. The experimental results with synthetic and real datasets (point cloud, temperature, and EEG data) demonstrate that our proposed method outperforms state-of-the-art methods.
The identification and modeling of time-varying systems is a fundamental challenge in signal processing and system identification. To address this challenge, we propose a class of time-varying state-space model (SSM) based neural networks in which the neurons' states are governed by time-varying dynamics. The proposed model provides the learnable time-varying dynamics through a dictionary of basis functions, where each basis function evolves differently over time. We evaluate the proposed approach on both synthetic data from switching systems and a speech denoising task where real audio is corrupted with switching dynamics noise. The results show that the proposed time-varying model consistently outperforms its time-invariant counterparts while maintaining comparable computational complexity. Our investigations also reveal which aspects of the time-varying dynamics of the data most need to be captured by the proposed time-invariant models, how the additional freedom provided by time-varying basis functions should be allocated across model components, and to what extent larger models can compensate for time-invariant limitations.
Training multimodal models to align time series with language runs into a self-supervision trap. The usual recipe asks an LLM to read a series and write a description, so label quality is capped by the perceptual skill the model is supposed to learn. The data can never teach more than the labeler already knows. A second gap makes this worse: most datasets use a single variable, but the patterns that matter (cross-channel correlation, lead-lag structure, co-occurring anomalies) appear only with several variables, right where the labeling LLM's limits are most exposed. These two problems create a trilemma: existing methods are reliable, realistic, or scalable, but none achieves all three. We resolve this by decoupling perception from description. Deterministic code computes a set of statistics from real, open-source multivariate series; the LLM verbalizes those precomputed facts. Perception, which LLMs do poorly, is handled by computation, while the LLM handles expression. This produces CGTime, our 4B-parameter computation-grounded time-series-language model. CGTime outperforms far larger general-purpose models on multivariate understanding tasks: it attains the best multivariate fact score on our held-out benchmark (0.283 vs. 0.173 for GPT-4o-mini and 0.203 for GPT-5.4-nano), a gap that survives Holm-corrected paired significance tests against every baseline. It also states verifiable numerical facts in generated captions more accurately and covers a broader range of statistical properties.