One can see deep-learning models as compositions of functions within the so-called tame geometry. In this expository note, we give an overview of some topics at the interface of tame geometry (also known as o-minimality), optimization theory, and deep learning theory and practice. To do so, we gradually introduce the concepts and tools used to build convergence guarantees for stochastic gradient descent in a general nonsmooth nonconvex, but tame, setting. This illustrates some ways in which tame geometry is a natural mathematical framework for the study of AI systems, especially within Deep Learning.

A B S T R A C T We study the Evolutionary Deep Neural Network (EDNN) framework for accelerating numerical solvers of time-dependent partial differential equations (PDEs). We introduce a Low-Rank Evolutionary Deep Neural Network (LR-EDNN), which constrains parameter evolution to a low-rank subspace, thereby reducing the effective dimensionality of training while preserving solution accuracy. The low-rank tangent subspace is defined layer-wise by the singular value decomposition (SVD) of the current network weights, and the resulting update is obtained by solving a well-posed, tractable linear system within this subspace. We evaluate LR-EDNN on representative PDE problems and compare it against corresponding baselines. Across cases, LR-EDNN achieves comparable accuracy with substantially fewer trainable parameters and reduced computational cost. These results indicate that low-rank constraints on parameter velocities, rather than full-space updates, provide a practical path toward scalable, efficient, and reproducible scientific machine learning for PDEs. Introduction The application of deep learning to solving partial differential equations (PDEs) has emerged as an active and promising research area, providing a powerful alternative to traditional numerical methods. Unlike classical approaches such as finite difference, finite element, or spectral methods, which rely on discretization and iterative solvers, deep learning methods leverage neural networks to approximate solutions directly, often bypassing the need for meshes and offering flexibility in handling irregular domains and high-dimensional problems [1, 2]. Early efforts in this domain focused primarily on two paradigms.

Manual HVAC fault diagnosis in commercial buildings takes 8-12 hours per incident and achieves only 60 percent diagnostic accuracy, reflecting analytics that stop at correlation instead of causation. To close this gap, we present GRID (Graph-based Reasoning for Intervention and Discovery), a three-stage causal discovery pipeline that combines constraint-based search, neural structural equation modeling, and language model priors to recover directed acyclic graphs from building sensor data. Across six benchmarks: synthetic rooms, EnergyPlus simulation, the ASHRAE Great Energy Predictor III dataset, and a live office testbed, GRID achieves F1 scores ranging from 0.65 to 1.00, with exact recovery (F1 = 1.00) in three controlled environments (Base, Hidden, Physical) and strong performance on real-world data (F1 = 0.89 on ASHRAE, 0.86 in noisy conditions). The method outperforms ten baseline approaches across all evaluation scenarios. Intervention scheduling achieves low operational impact in most scenarios (cost <= 0.026) while reducing risk metrics compared to baseline approaches. The framework integrates constraint-based methods, neural architectures, and domain-specific language model prompts to address the observational-causal gap in building analytics.

Developing trustworthy intelligent vision systems for high-stakes domains, \emph{e.g.}, remote sensing and medical diagnosis, demands broad robustness without costly retraining. We propose \textbf{Visual Reasoning Agent (VRA)}, a training-free, agentic reasoning framework that wraps off-the-shelf vision-language models \emph{and} pure vision systems in a \emph{Think--Critique--Act} loop. While VRA incurs significant additional test-time computation, it achieves up to 40\% absolute accuracy gains on challenging visual reasoning benchmarks. Future work will optimize query routing and early stopping to reduce inference overhead while preserving reliability in vision tasks.

This study deploys methods in explainable artificial intelligence (XAI), e.g. decision trees and Pearson's correlation coefficient (PCC), to investigate electricity usage in multiple locales. It addresses the vital issue of energy burden, i.e. total amount spent on energy divided by median household income. Socio-demographic data is analyzed with energy features, especially using decision trees and PCC, providing explainable predictors on factors affecting energy burden. Based on the results of the analysis, a pilot energy equity web portal is designed along with a novel energy burden calculator. Leveraging XAI, this portal (with its calculator) serves as a prototype information system that can offer tailored actionable advice to multiple energy stakeholders. The ultimate goal of this study is to promote greater energy equity through the adaptation of XAI methods for energy-related analysis with suitable recommendations.

We present EPIC, an AI-driven platform designed to augment operational data analytics. EPIC employs a hierarchical multi-agent architecture where a top-level large language model provides query processing, reasoning and synthesis capabilities. These capabilities orchestrate three specialized low-level agents for information retrieval, descriptive analytics, and predictive analytics. This architecture enables EPIC to perform HPC operational analytics on multi-modal data, including text, images, and tabular formats, dynamically and iteratively. EPIC addresses the limitations of existing HPC operational analytics approaches, which rely on static methods that struggle to adapt to evolving analytics tasks and stakeholder demands. Through extensive evaluations on the Frontier HPC system, we demonstrate that EPIC effectively handles complex queries. Using descriptive analytics as a use case, fine-tuned smaller models outperform large state-of-the-art foundation models, achieving up to 26% higher accuracy. Additionally, we achieved 19x savings in LLM operational costs compared to proprietary solutions by employing a hybrid approach that combines large foundational models with fine-tuned local open-weight models.

--Transient stability prediction is critically essential to the fast online assessment and maintaining the stable operation in power systems. The wide deployment of phasor measurement units (PMUs) promotes the development of data-driven approaches for transient stability assessment. This paper proposes the temporal and topological embedding deep neural network (TTEDNN) model to forecast transient stability with the early transient dynamics. The TTEDNN model can accurately and efficiently predict the transient stability by extracting the temporal and topological features from the time-series data of the early transient dynamics. The grid-informed adjacency matrix is used to incorporate the power grid structural and electrical parameter information. The transient dynamics simulation environments under the single-node and multiple-node perturbations are used to test the performance of the TTEDNN model for the IEEE 39-bus and IEEE 118-bus power systems. The results show that the TTEDNN model has the best and most robust prediction performance. Furthermore, the TTEDNN model also demonstrates the transfer capability to predict the transient stability in the more complicated transient dynamics simulation environments.

Decentralized combinatorial optimization in evolving multi-agent systems poses significant challenges, requiring agents to balance long-term decision-making, short-term optimized collective outcomes, while preserving autonomy of interactive agents under unanticipated changes. Reinforcement learning offers a way to model sequential decision-making through dynamic programming to anticipate future environmental changes. However, applying multi-agent reinforcement learning (MARL) to decentralized combinatorial optimization problems remains an open challenge due to the exponential growth of the joint state-action space, high communication overhead, and privacy concerns in centralized training. To address these limitations, this paper proposes Hierarchical Reinforcement and Collective Learning (HRCL), a novel approach that leverages both MARL and decentralized collective learning based on a hierarchical framework. Agents take high-level strategies using MARL to group possible plans for action space reduction and constrain the agent behavior for Pareto optimality. Meanwhile, the low-level collective learning layer ensures efficient and decentralized coordinated decisions among agents with minimal communication. Extensive experiments in a synthetic scenario and real-world smart city application models, including energy self-management and drone swarm sensing, demonstrate that HRCL significantly improves performance, scalability, and adaptability compared to the standalone MARL and collective learning approaches, achieving a win-win synthesis solution.

Abstract--End-to-end reinforcement learning (RL) for humanoid locomotion is appealing for its compact perception-action mapping, yet practical policies often suffer from training instability, inefficient feature fusion, and high actuation cost. We present HuMam, a state-centric end-to-end RL framework that employs a single-layer Mamba encoder to fuse robot-centric states with oriented footstep targets and a continuous phase clock. The policy outputs joint position targets tracked by a low-level PD loop and is optimized with PPO. On the JVRC-1 humanoid in mc-mujoco, HuMam consistently improves learning efficiency, training stability, and overall task performance over a strong feedforward baseline, while reducing power consumption and torque peaks. T o our knowledge, this is the first end-to-end humanoid RL controller that adopts Mamba as the fusion backbone, demonstrating tangible gains in efficiency, stability, and control economy. UMANOID locomotion demands controllers that are both foresightful and resource-aware: foresightful to coordinate accurate foot placement and whole-body balance, and resource-aware to run reliably under onboard compute and actuation limits [1]. End-to-end reinforcement learning (RL) is attractive because it can discover feedback strategies directly from interaction [2]; however, its effectiveness hinges on (i) how heterogeneous inputs are fused and (ii) how training is shaped to avoid trivial or unstable behaviors.

Combining multimodal data is a key issue in a wide range of machine learning tasks, including many remote sensing problems. In Earth observation, early multimodal data fusion methods were based on specific neural network architectures and supervised learning. Ever since, the scarcity of labeled data has motivated self-supervised learning techniques. State-of-the-art multimodal representation learning techniques leverage the spatial alignment between satellite data from different modalities acquired over the same geographic area in order to foster a semantic alignment in the latent space. In this paper, we investigate how this methods can preserve task-relevant information that is not shared across modalities. First, we show, under simplifying assumptions, when alignment strategies fundamentally lead to an information loss. Then, we support our theoretical insight through numerical experiments in more realistic settings. With those theoretical and empirical evidences, we hope to support new developments in contrastive learning for the combination of multimodal satellite data. Our code and data is publicly available at https://github.com/Romain3Ch216/alg_maclean_25.