From Classical to Topological Neural Networks Under Uncertainty

This chapter explores neural networks, topological data analysis, and topological deep learning techniques, alongside statistical Bayesian methods, for processing images, time series, and graphs to maximize the potential of artificial intelligence in the military domain. Throughout the chapter, we highlight practical applications spanning image, video, audio, and time-series recognition, fraud detection, and link prediction for graphical data, illustrating how topology-aware and uncertainty-aware models can enhance robustness, interpretability, and generalization.

Low Rank Transformer for Multivariate Time Series Anomaly Detection and Localization

Multivariate time series (MTS) anomaly diagnosis, which encompasses both anomaly detection and localization, is critical for the safety and reliability of complex, large-scale real-world systems. The vast majority of existing anomaly diagnosis methods offer limited theoretical insights, especially for anomaly localization, which is a vital but largely unexplored area. The aim of this contribution is to study the learning process of a Transformer when applied to MTS by revealing connections to statistical time series methods. Based on these theoretical insights, we propose the Attention Low-Rank Transformer (ALoRa-T) model, which applies low-rank regularization to self-attention, and we introduce the Attention Low-Rank score, effectively capturing the temporal characteristics of anomalies. Finally, to enable anomaly localization, we propose the ALoRa-Loc method, a novel approach that associates anomalies to specific variables by quantifying interrelationships among time series. Extensive experiments and real data analysis, show that the proposed methodology significantly outperforms state-of-the-art methods in both detection and localization tasks.

Continuous-Time Analysis of AFDM: Fundamental Bounds, and Effects of Pulse-Shaping and Hardware Impairments

Affine frequency division multiplexing (AFDM) has recently emerged as a resilient waveform candidate for high-mobility next-generation wireless systems. However, current literature mostly focuses on discrete time (DT) models, often overlooking effects and hardware non-idealities of actual continuous time (CT) signal generation. In this paper, we bridge this gap by developing a CT-analytical framework based on the affine Fourier series (AFS) representation, which allows us to demonstrate that strictly bandlimited pulses and subcarrier suppression strategies are essential to maintain the multicarrier structure of the signal. In addition, we derive the analytical power spectral density of AFDM and evaluate its spectral characteristics in comparison with other multicarrier schemes, considering the impact of realistic truncated pulse-shaping. Furthermore, we analyze the sensitivity of the CT model to phase noise, carrier frequency offset, and sampling jitter, providing a theoretical analysis of communication performance. Finally, we derive closed-form Cramér-Rao bounds for channel parameter estimation, showing that the chirped modulation peculiar of AFDM increases the estimation variance but enables the resolution of Doppler ambiguities. Our findings provide the necessary theoretical and practical foundations for the implementation of AFDM in realistic wireless transceivers.

Tree crop mapping of South America reveals links to deforestation and conservation

Monitoring tree crop expansion is vital for zero-deforestation policies like the European Union's Regulation on Deforestation-free Products (EUDR). However, these efforts are hindered by a lack of highresolution data distinguishing diverse agricultural systems from forests. Here, we present the first 10m-resolution tree crop map for South America, generated using a multi-modal, spatio-temporal deep learning model trained on Sentinel-1 and Sentinel-2 satellite imagery time series. The map identifies approximately 11 million hectares of tree crops, 23% of which is linked to 2000-2020 forest cover loss. Critically, our analysis reveals that existing regulatory maps supporting the EUDR often classify established agriculture, particularly smallholder agroforestry, as "forest". This discrepancy risks false deforestation alerts and unfair penalties for small-scale farmers. Our work mitigates this risk by providing a high-resolution baseline, supporting conservation policies that are effective, inclusive, and equitable.

Adjacency Spectral Embeddings of Correlation Networks

In many applications, weighted networks are constructed based on time series data: each time series is associated to a vertex and edge weights are given by pairwise correlations. The result is a network whose edge dependency structure violates the assumptions of most common network models. Nonetheless, it is common to analyze these "correlation networks" using embedding methods derived from edge-independent network models, based on a belief that the edges are approximately independent. In this work, we put this modeling choice on firm theoretical ground. We show that when the time series are expressible in terms of a small number of Fourier basis elements (or in some other suitably-chosen basis), correlation networks correspond to latent space networks with dependent edge noise in which the vertex-level latent variables encode the basis coefficients. Further, we show that when time series are observed subject to noise, spectral embedding of the resulting noisy correlation network still recovers these true vertex-level latent representations under suitable assumptions. This characterization of embeddings as learning Fourier coefficients appears to be folklore in the signal processing community in the context of principal component analysis, but is, to the best of our knowledge, new to the statistical network analysis literature.

Augmenting representations with scientific papers

Astronomers have acquired vast repositories of multimodal data, including images, spectra, and time series, complemented by decades of literature that analyzes astrophysical sources. Still, these data sources are rarely systematically integrated. This work introduces a contrastive learning framework designed to align X-ray spectra with domain knowledge extracted from scientific literature, facilitating the development of shared multimodal representations. Establishing this connection is inherently complex, as scientific texts encompass a broader and more diverse physical context than spectra. We propose a contrastive pipeline that achieves a 20% Recall@1% when retrieving texts from spectra, proving that a meaningful alignment between these modalities is not only possible but capable of accelerating the interpretation of rare or poorly understood sources. Furthermore, the resulting shared latent space effectively encodes physically significant information. By fusing spectral and textual data, we improve the estimation of 20 physical variables by 16-18% over unimodal spectral baselines. Our results indicate that a Mixture of Experts (MoE) strategy, which leverages both unimodal and shared representations, yields superior performance. Finally, outlier analysis within the multimodal latent space identifies high-priority targets for follow-up investigation, including a candidate pulsating ULX (PULX) and a gravitational lens system. Importantly, this framework can be extended to other scientific domains where aligning observational data with existing literature is possible.

Stringology-Based Motif Discovery from EEG Signals: an ADHD Case Study

We propose a novel computational framework for analyzing electroencephalography (EEG) time series using methods from stringology, the study of efficient algorithms for string processing, to systematically identify and characterize recurrent temporal patterns in neural signals. The primary aim is to introduce quantitative measures to understand neural signal dynamics, with the present findings serving as a proof-of-concept. The framework adapts order-preserving matching (OPM) and Cartesian tree matching (CTM) to detect temporal motifs that preserve relative ordering and hierarchical structure while remaining invariant to amplitude scaling. This approach provides a temporally precise representation of EEG dynamics that complements traditional spectral and global complexity analyses. To evaluate its utility, we applied the framework to multichannel EEG recordings from individuals with attention-deficit/hyperactivity disorder (ADHD) and matched controls using a publicly available dataset. Highly recurrent, group-specific motifs were extracted and quantified using both OPM and CTM. The ADHD group exhibited significantly higher motif frequencies, suggesting increased repetitiveness in neural activity. OPM analysis revealed shorter motif lengths and greater gradient instability in ADHD, reflected in larger mean and maximal inter-sample amplitude changes. CTM analysis further demonstrated reduced hierarchical complexity in ADHD, characterized by shallower tree structures and fewer hierarchical levels despite comparable motif lengths. These findings suggest that ADHD-related EEG alterations involve systematic differences in the structure, stability, and hierarchical organization of recurrent temporal patterns. The proposed stringology-based motif framework provides a complementary computational tool with potential applications for objective biomarker development in neurodevelopmental disorders.

It's TIME: Towards the Next Generation of Time Series Forecasting Benchmarks

Time series foundation models (TSFMs) are revolutionizing the forecasting landscape from specific dataset modeling to generalizable task evaluation. However, we contend that existing benchmarks exhibit common limitations in four dimensions: constrained data composition dominated by reused legacy sources, compromised data integrity lacking rigorous quality assurance, misaligned task formulations detached from real-world contexts, and rigid analysis perspectives that obscure generalizable insights. To bridge these gaps, we introduce TIME, a next-generation task-centric benchmark comprising 50 fresh datasets and 98 forecasting tasks, tailored for strict zero-shot TSFM evaluation free from data leakage. Integrating large language models and human expertise, we establish a rigorous human-in-the-loop benchmark construction pipeline to ensure high data integrity and redefine task formulation by aligning forecasting configurations with real-world operational requirements and variate predictability. Furthermore, we propose a novel pattern-level evaluation perspective that moves beyond traditional dataset-level evaluations based on static meta labels. By leveraging structural time series features to characterize intrinsic temporal properties, this approach offers generalizable insights into model capabilities across diverse patterns. We evaluate 12 representative TSFMs and establish a multi-granular leaderboard to facilitate in-depth analysis and visualized inspection. The leaderboard is available at https://huggingface.co/spaces/Real-TSF/TIME-leaderboard.

Continuous-Time Piecewise-Linear Recurrent Neural Networks

In dynamical systems reconstruction (DSR) we aim to recover the dynamical system (DS) underlying observed time series. Specifically, we aim to learn a generative surrogate model which approximates the underlying, data-generating DS, and recreates its long-term properties (`climate statistics'). In scientific and medical areas, in particular, these models need to be mechanistically tractable -- through their mathematical analysis we would like to obtain insight into the recovered system's workings. Piecewise-linear (PL), ReLU-based RNNs (PLRNNs) have a strong track-record in this regard, representing SOTA DSR models while allowing mathematical insight by virtue of their PL design. However, all current PLRNN variants are discrete-time maps. This is in disaccord with the assumed continuous-time nature of most physical and biological processes, and makes it hard to accommodate data arriving at irregular temporal intervals. Neural ODEs are one solution, but they do not reach the DSR performance of PLRNNs and often lack their tractability. Here we develop theory for continuous-time PLRNNs (cPLRNNs): We present a novel algorithm for training and simulating such models, bypassing numerical integration by efficiently exploiting their PL structure. We further demonstrate how important topological objects like equilibria or limit cycles can be determined semi-analytically in trained models. We compare cPLRNNs to both their discrete-time cousins as well as Neural ODEs on DSR benchmarks, including systems with discontinuities which come with hard thresholds.

Power Interpretable Causal ODE Networks: A Unified Model for Explainable Anomaly Detection and Root Cause Analysis in Power Systems

Anomaly detection and root cause analysis (RCA) are critical for ensuring the safety and resilience of cyber-physical systems such as power grids. However, existing machine learning models for time series anomaly detection often operate as black boxes, offering only binary outputs without any explanation, such as identifying anomaly type and origin. To address this challenge, we propose Power Interpretable Causality Ordinary Differential Equation (PICODE) Networks, a unified, causality-informed architecture that jointly performs anomaly detection along with the explanation why it is detected as an anomaly, including root cause localization, anomaly type classification, and anomaly shape characterization. Experimental results in power systems demonstrate that PICODE achieves competitive detection performance while offering improved interpretability and reduced reliance on labeled data or external causal graphs. We provide theoretical results demonstrating the alignment between the shape of anomaly functions and the changes in the weights of the extracted causal graphs.