The twenty-first century is a data-driven era where human activities and behavior, physical phenomena, scientific discoveries, technology advancements, and almost everything that happens in the world resulting in massive generation, collection, and utilization of data. Connectivity in data is a crucial property. A straightforward example is the World Wide Web, where every webpage is connected to other web pages through hyperlinks, providing a form of directed connectivity. Combinatorial data refers to combinations of data items based on certain connectivity rules. Other forms of combinatorial data include social networks, meshes, community clusters, set systems, and molecules. This Ph.D. dissertation focuses on learning and computing with combinatorial data. We study and examine topological and connectivity features within and across connected data to improve the performance of learning and achieve high algorithmic efficiency.

Artificial intelligence commonly refers to the science and engineering of artificial systems that can carry out tasks generally associated with requiring aspects of human intelligence, such as playing games, translating languages, and driving cars. In recent years, there have been exciting advances in learning-based, data-driven approaches towards AI, and machine learning and deep learning have enabled computer systems to perceive the world in unprecedented ways. Reinforcement learning has enabled breakthroughs in complex games such as Go and challenging robotics tasks such as quadrupedal locomotion. A key aspect of intelligence is to not only make predictions, but reason about the uncertainty in these predictions, and to consider this uncertainty when making decisions. This is what this manuscript on "Probabilistic Artificial Intelligence" is about. The first part covers probabilistic approaches to machine learning. We discuss the differentiation between "epistemic" uncertainty due to lack of data and "aleatoric" uncertainty, which is irreducible and stems, e.g., from noisy observations and outcomes. We discuss concrete approaches towards probabilistic inference and modern approaches to efficient approximate inference. The second part of the manuscript is about taking uncertainty into account in sequential decision tasks. We consider active learning and Bayesian optimization -- approaches that collect data by proposing experiments that are informative for reducing the epistemic uncertainty. We then consider reinforcement learning and modern deep RL approaches that use neural network function approximation. We close by discussing modern approaches in model-based RL, which harness epistemic and aleatoric uncertainty to guide exploration, while also reasoning about safety.

The attention mechanism was originally introduced in the context of large language models to learn relations between words [26]. Solving the many-electron Schrödinger equation for Here, the attention mechanism is employed to identify solids is an exceedingly difficult problem due to the exponential and quantify how electrons influence each other and how growth of the Hilbert space dimension. Various such influence affects their individual orbitals. This enable techniques based on the variational principle have long the construction of NN wavefunctions from Slater been developed to approximate the ground state of interacting determinants of generalized orbitals that depend on the electron systems using trial wavefunctions.

A generative AI architecture called bubbly flow generative adversarial networks (BF-GAN) is developed, designed to generate realistic and high-quality bubbly flow images through physically conditioned inputs, jg and jf. Initially, 52 sets of bubbly flow experiments under varying conditions are conducted to collect 140,000 bubbly flow images with physical labels of jg and jf for training data. A multi-scale loss function is then developed, incorporating mismatch loss and pixel loss to enhance the generative performance of BF-GAN further. Regarding evaluative metrics of generative AI, the BF-GAN has surpassed conventional GAN. Physically, key parameters of bubbly flow generated by BF-GAN are extracted and compared with measurement values and empirical correlations, validating BF-GAN's generative performance. The comparative analysis demonstrate that the BF-GAN can generate realistic and high-quality bubbly flow images with any given jg and jf within the research scope. BF-GAN offers a generative AI solution for two-phase flow research, substantially lowering the time and cost required to obtain high-quality data. In addition, it can function as a benchmark dataset generator for bubbly flow detection and segmentation algorithms, enhancing overall productivity in this research domain. The BF-GAN model is available online (https://github.com/zhouzhouwen/BF-GAN).

Estimation and inference for the Average Treatment Effect (ATE) is a cornerstone of causal inference and often serves as the foundation for developing procedures for more complicated settings. Although traditionally analyzed in a batch setting, recent advances in martingale theory have paved the way for adaptive methods that can enhance the power of downstream inference. Despite these advances, progress in understanding and developing adaptive algorithms remains in its early stages. Existing work either focus on asymptotic analyses that overlook exploration-exploitation tradeoffs relevant in finite-sample regimes or rely on simpler but suboptimal estimators. In this work, we address these limitations by studying adaptive sampling procedures that take advantage of the asymptotically optimal Augmented Inverse Probability Weighting (AIPW) estimator. Our analysis uncovers challenges obscured by asymptotic approaches and introduces a novel algorithmic design principle reminiscent of optimism in multiarmed bandits. This principled approach enables our algorithm to achieve significant theoretical and empirical gains compared to prior methods. Our findings mark a step forward in advancing adaptive causal inference methods in theory and practice.

Turbulent flow fields are characterized by extreme events that are statistically intermittent and carry a significant amount of energy and physical importance. To emulate these flows, we introduce the extreme variational Autoencoder (xVAE), which embeds a max-infinitely divisible process with heavy-tailed distributions into a standard VAE framework, enabling accurate modeling of extreme events. xVAEs are neural network models that reduce system dimensionality by learning non-linear latent representations of data. We demonstrate the effectiveness of xVAE in large-eddy simulation data of wildland fire plumes, where intense heat release and complex plume-atmosphere interactions generate extreme turbulence. Comparisons with the commonly used Proper Orthogonal Decomposition (POD) modes show that xVAE is more robust in capturing extreme values and provides a powerful uncertainty quantification framework using variational Bayes. Additionally, xVAE enables analysis of the so-called copulas of fields to assess risks associated with rare events while rigorously accounting for uncertainty, such as simultaneous exceedances of high thresholds across multiple locations. The proposed approach provides a new direction for studying realistic turbulent flows, such as high-speed aerodynamics, space propulsion, and atmospheric and oceanic systems that are characterized by extreme events.

Bivariate structural causal models (SCM) are often used to infer causal direction by examining their goodness-of-fit under restricted model classes. In this paper, we describe a parametrization of bivariate SCMs in terms of a causal velocity by viewing the cause variable as time in a dynamical system. The velocity implicitly defines counterfactual curves via the solution of initial value problems where the observation specifies the initial condition. Using tools from measure transport, we obtain a unique correspondence between SCMs and the score function of the generated distribution via its causal velocity. Based on this, we derive an objective function that directly regresses the velocity against the score function, the latter of which can be estimated non-parametrically from observational data. We use this to develop a method for bivariate causal discovery that extends beyond known model classes such as additive or location scale noise, and that requires no assumptions on the noise distributions. When the score is estimated well, the objective is also useful for detecting model non-identifiability and misspecification. We present positive results in simulation and benchmark experiments where many existing methods fail, and perform ablation studies to examine the method's sensitivity to accurate score estimation.

Abstract--Accurate estimation of voltage drop (IR drop) in modern Application-Specific Integrated Circuits (ASICs) is highly time and resource demanding, due to the growing complexity and the transistor density in recent technology nodes. To mitigate this challenge, we investigate how Machine Learning (ML) techniques, including Extreme Gradient Boosting (XGBoost), Convolutional Neural Network (CNN), and Graph Neural Network (GNN) can aid in reducing the computational effort and implicitly the time required to estimate the IR drop in Integrated Circuits (ICs). ML algorithms, on the other hand, are explored as an alternative solution to offer quick and precise IR drop estimation, but in considerably less time. This study illustrates the effectiveness of ML algorithms in precisely estimating IR drop and optimizing ASIC sign-off. Thus, a new round of simulations is required for verification. This process is a standard routine in every ASIC design and manufacturing process, and it is defined as the "sign-off" REDICTION of IR drop is an important problem faced today often by ASIC designers. As the current (I) flows With the transition to larger density integration of transistors, through the Power Distribution Network (PDN), a part of the number of connection layers and interconnections the applied voltage inherently drops across the current path, have increased exponentially over the last decades, driven which is, in simple terms, the definition of IR drop. As a result, while commercial results in voltage drop, or to the grounding (GND), which tools are trying to keep up with the up-scaling demand, results in a ground bounce.

AFoundation Models (FM) have increasingly drawn the attention of researchers due to their scalability and generalization across diverse tasks. Inspired by the success of FMs and the principles that have driven advancements in Large Language Models (LLMs), we introduce MoFM as a novel Motion Foundation Model. MoFM is designed for the semantic understanding of complex human motions in both time and space. To facilitate large-scale training, MotionBook, a comprehensive human motion dictionary of discretized motions is designed and employed. MotionBook utilizes Thermal Cubes to capture spatio-temporal motion heatmaps, applying principles from discrete variational models to encode human movements into discrete units for a more efficient and scalable representation. MoFM, trained on a large corpus of motion data, provides a foundational backbone adaptable to diverse downstream tasks, supporting paradigms such as one-shot, unsupervised, and supervised tasks. This versatility makes MoFM well-suited for a wide range of motion-based applications.

The increased availability of data and computing resources has enabled researchers to successfully adopt machine learning (ML) techniques and make significant contributions in several engineering areas. ML and in particular deep learning (DL) algorithms have shown to perform better in tasks where a physical bottom-up description of the phenomenon is lacking and/or is mathematically intractable. Indeed, they take advantage of the observations of natural phenomena to automatically acquire knowledge and learn internal relations. Despite the historical model-based mindset, communications engineering recently started shifting the focus towards top-down data-driven learning models, especially in domains such as channel modeling and physical layer design, where in most of the cases no general optimal strategies are known. In this thesis, we aim at solving some fundamental open challenges in physical layer communications exploiting new DL paradigms. In particular, we mathematically formulate, under ML terms, classic problems such as channel capacity and optimal coding-decoding schemes, for any arbitrary communication medium. We design and develop the architecture, algorithm and code necessary to train the equivalent DL model, and finally, we propose novel solutions to long-standing problems in the field.