Many tasks require mapping continuous input data (e.g. images) to discrete task outputs (e.g. class labels). Yet, how neural networks learn to perform such discrete computations on continuous data manifolds remains poorly understood. Here, we show that signatures of such computations emerge in the representational geometry of neural networks as they learn. By analysing the Riemannian pullback metric across layers of a neural network, we find that network computation can be decomposed into two functions: discretising continuous input features and performing logical operations on these discretised variables. Furthermore, we demonstrate how different learning regimes (rich vs. lazy) have contrasting metric and curvature structures, affecting the ability of the networks to generalise to unseen inputs. Overall, our work provides a geometric framework for understanding how neural networks learn to perform discrete computations on continuous manifolds.

We study streaming data with categorical features where the vocabulary of categorical feature values is changing and can even grow unboundedly over time. Feature hashing is commonly used as a pre-processing step to map these categorical values into a feature space of fixed size before learning their embeddings. While these methods have been developed and evaluated for offline or batch settings, in this paper we consider online settings. We show that deterministic embeddings are sensitive to the arrival order of categories and suffer from forgetting in online learning, leading to performance deterioration. To mitigate this issue, we propose a probabilistic hash embedding (PHE) model that treats hash embeddings as stochastic and applies Bayesian online learning to learn incrementally from data. Based on the structure of PHE, we derive a scalable inference algorithm to learn model parameters and infer/update the posteriors of hash embeddings and other latent variables. Our algorithm (i) can handle an evolving vocabulary of categorical items, (ii) is adaptive to new items without forgetting old items, (iii) is implementable with a bounded set of parameters that does not grow with the number of distinct observed values on the stream, and (iv) is invariant to the item arrival order. Experiments in classification, sequence modeling, and recommendation systems in online learning setups demonstrate the superior performance of PHE while maintaining high memory efficiency (consumes as low as 2~4 memory of a one-hot embedding table). Supplementary materials are at https://github.com/aodongli/probabilistic-hash-embeddings

Stochastic Variance Reduced Gradient (SVRG) and its variants aim to speed-up training by using gradient corrections, but have seen limited success in deep learning. Here, we show surprising new foundational connections of SVRG to a recently proposed Bayesian method called posterior correction. Specifically, we show that SVRG is recovered as a special case of posterior correction over the isotropic-Gaussian family, while novel extensions are automatically obtained by using more flexible exponential families. We derive two new SVRG variants by using Gaussian families: First, a Newton-like variant that employs novel Hessian corrections, and second, an Adam-like extension that improves pretraining and finetuning of Transformer language models. This is the first work to connect SVRG to Bayes and use it to boost variational training for deep networks.

Modeling historical events poses fundamental challenges for machine learning: extreme data scarcity (N << 100), heterogeneous and noisy measurements, missing counterfactuals, and the requirement for human interpretable explanations. We present HistoricalML, a probabilistic neuro-symbolic framework that addresses these challenges through principled integration of (1) Bayesian uncertainty quantification to separate epistemic from aleatoric uncertainty, (2) structural causal models for counterfactual reasoning under confounding, (3) cooperative game theory (Shapley values) for fair allocation modeling, and (4) attention based neural architectures for context dependent factor weighting. We provide theoretical analysis showing that our approach achieves consistent estimation in the sparse data regime when strong priors from domain knowledge are available, and that Shapley based allocation satisfies axiomatic fairness guarantees that pure regression approaches cannot provide. We instantiate the framework on two historical case studies: the 19th century partition of Africa (N = 7 colonial powers) and the Second Punic War (N = 2 factions). Our model identifies Germany's +107.9 percent discrepancy as a quantifiable structural tension preceding World War I, with tension factor 36.43 and 0.79 naval arms race correlation. For the Punic Wars, Monte Carlo battle simulations achieve a 57.3 percent win probability for Carthage at Cannae and 57.8 percent for Rome at Zama, aligning with historical outcomes. Counterfactual analysis reveals that Carthaginian political support (support score 6.4 vs Napoleon's 7.1), rather than military capability, was the decisive factor.

Collaborative Combat Aircraft (CCAs) are envisioned to enable autonomous Intelligence, Surveillance, and Reconnaissance (ISR) missions in contested environments, where adversaries may act strategically to deceive or evade detection. These missions pose challenges due to model uncertainty and the need for safe, real-time decision-making. Robust Markov Decision Processes (RMDPs) provide worst-case guarantees but are limited by static ambiguity sets that capture initial uncertainty without adapting to new observations. This paper presents an adaptive RMDP framework tailored to ISR missions with CCAs. We introduce a mission-specific formulation in which aircraft alternate between movement and sensing states. Adversarial tactics are modeled as a finite set of transition kernels, each capturing assumptions about how adversarial sensing or environmental conditions affect rewards. Our approach incrementally refines policies by eliminating inconsistent threat models, allowing agents to shift from conservative to aggressive behaviors while maintaining robustness. We provide theoretical guarantees showing that the adaptive planner converges as credible sets contract to the true threat and maintains safety under uncertainty. Experiments under Gaussian and non-Gaussian threat models across diverse network topologies show higher mission rewards and fewer exposure events compared to nominal and static robust planners.

Bayesian Neural Networks provide a principled framework for uncertainty quantification by modeling the posterior distribution of network parameters. However, exact posterior inference is computationally intractable, and widely used approximations like the Laplace method struggle with scalability and posterior accuracy in modern deep networks. In this work, we revisit sampling techniques for posterior exploration, proposing a simple variation tailored to efficiently sample from the posterior in over-parameterized networks by leveraging the low-dimensional structure of loss minima. Building on this, we introduce a model that learns a deformation of the parameter space, enabling rapid posterior sampling without requiring iterative methods. Empirical results demonstrate that our approach achieves competitive posterior approximations with improved scalability compared to recent refinement techniques. These contributions provide a practical alternative for Bayesian inference in deep learning.

Quantifying uncertainty in deep regression models is important both for understanding the confidence of the model and for safe decision-making in high-risk domains. Existing approaches that yield prediction intervals overlook distributional information, neglecting the effect of multimodal or asymmetric distributions on decision-making. Similarly, full or approximated Bayesian methods, while yielding the predictive posterior density, demand major modifications to the model architecture and retraining. We introduce MCNF, a novel post hoc uncertainty quantification method that produces both prediction intervals and the full conditioned predictive distribution. MCNF operates on top of the underlying trained predictive model; thus, no predictive model retraining is needed. We provide experimental evidence that the MCNF-based uncertainty estimate is well calibrated, is competitive with state-of-the-art uncertainty quantification methods, and provides richer information for downstream decision-making tasks.

Random telegraph noise is a prevalent variability phenomenon in nanoelectronic devices, arising from stochastic carrier exchange at defect sites and critically impacting device reliability and performance. Conventional analysis techniques often rely on restrictive assumptions or manual interventions, limiting their applicability to complex, noisy datasets. Here, we introduce RTNinja, a generalized, fully automated machine learning framework for the unsupervised analysis of random telegraph noise signals. RTNinja deconvolves complex signals to identify the number and characteristics of hidden individual sources without requiring prior knowledge of the system. The framework comprises two modular components: LevelsExtractor, which uses Bayesian inference and model selection to denoise and discretize the signal, and SourcesMapper, which infers source configurations through probabilistic clustering and optimization. To evaluate performance, we developed a Monte Carlo simulator that generates labeled datasets spanning broad signal-to-noise ratios and source complexities; across 7000 such datasets, RTNinja consistently demonstrated high-fidelity signal reconstruction and accurate extraction of source amplitudes and activity patterns. Our results demonstrate that RTNinja offers a robust, scalable, and device-agnostic tool for random telegraph noise characterization, enabling large-scale statistical benchmarking, reliability-centric technology qualification, predictive failure modeling, and device physics exploration in next-generation nanoelectronics.

Abstract--Training effective artificial intelligence models for telecommunications is challenging due to the scarcity of deployment-specific data. Real data collection is expensive, and available datasets often fail to capture the unique operational conditions and contextual variability of the network environment. Digital twinning provides a potential solution to this problem, as simulators tailored to the current network deployment can generate site-specific data to augment the available training datasets. However, there is a need to develop solutions to bridge the inherent simulation-to-reality (sim-to-real) gap between synthetic and real-world data. This paper reviews recent advances on two complementary strategies: 1) the calibration of digital twins (DTs) through real-world measurements, and 2) the use of sim-to-real gap-aware training strategies to robustly handle residual discrepancies between digital twin-generated and real data. For the latter, we evaluate two conceptually distinct methods that model the sim-to-real gap either at the level of the environment via Bayesian learning or at the level of the training loss via prediction-powered inference. Driven by the continued growth of computing resources and training datasets, artificial intelligence (AI) research is widely considered to be in the scaling era, which is focused on the development of general-purpose models that exhibit emergent capabilities. While this trend has yielded impressive results for many tasks, particularly in the domain of language modeling, it poses unique challenges when applied to engineering domains such as telecommunication networks.

Generation IV (Gen-IV) nuclear power plants are envisioned to replace the current reactor fleet, bringing improvements in performance, safety, reliability, and sustainability. However, large cost investments currently inhibit the deployment of these advanced reactor concepts. Digital twins bridge real-world systems with digital tools to reduce costs, enhance decision-making, and boost operational efficiency. In this work, a digital twin framework is designed to operate the Gen-IV Fluoride-salt-cooled High-temperature Reactor, utilizing data-enhanced methods to optimize operational and maintenance policies while adhering to system constraints. The closed-loop framework integrates surrogate modeling, reinforcement learning, and Bayesian inference to streamline end-to-end communication for online regulation and self-adjustment. Reinforcement learning is used to consider component health and degradation to drive the target power generations, with constraints enforced through a Reference Governor control algorithm that ensures compliance with pump flow rate and temperature limits. These input driving modules benefit from detailed online simulations that are assimilated to measurement data with Bayesian filtering. The digital twin is demonstrated in three case studies: a one-year long-term operational period showcasing maintenance planning capabilities, short-term accuracy refinement with high-frequency measurements, and system shock capturing that demonstrates real-time recalibration capabilities when change in boundary conditions. These demonstrations validate robustness for health-aware and constraint-informed nuclear plant operation, with general applicability to other advanced reactor concepts and complex engineering systems.