Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation constraint from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By optimizing over the reproducing kernel Hilbert space norm while applying a maximum likelihood estimation penalty on kernel complexity, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface fracture networks and arterial blood flow. Our results show that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical.

Understanding the complex interactions within dynamic multilayer networks is critical for advancements in various scientific domains. Existing models often fail to capture such networks' temporal and cross-layer dynamics. This paper introduces a novel Tensor State Space Model for Dynamic Multilayer Networks (TSSDMN), utilizing a latent space model framework. TSSDMN employs a symmetric Tucker decomposition to represent latent node features, their interaction patterns, and layer transitions. Then by fixing the latent features and allowing the interaction patterns to evolve over time, TSSDMN uniquely captures both the temporal dynamics within layers and across different layers. The model identifiability conditions are discussed. By treating latent features as variables whose posterior distributions are approximated using a mean-field variational inference approach, a variational Expectation Maximization algorithm is developed for efficient model inference. Numerical simulations and case studies demonstrate the efficacy of TSSDMN for understanding dynamic multilayer networks.

The importance of clinical variables in the prognosis of the disease is explained using statistical correlation or machine learning (ML). However, the predictive importance of these variables may not represent their causal relationships with diseases. This paper uses clinical variables from a heart failure (HF) patient cohort to investigate the causal explainability of important variables obtained in statistical and ML contexts. Due to inherent regression modeling, popular causal discovery methods strictly assume that the cause and effect variables are numerical and continuous. This paper proposes a new computational framework to enable causal structure discovery (CSD) and score the causal strength of mixed-type (categorical, numerical, binary) clinical variables for binary disease outcomes. In HF classification, we investigate the association between the importance rank order of three feature types: correlated features, features important for ML predictions, and causal features. Our results demonstrate that CSD modeling for nonlinear causal relationships is more meaningful than its linear counterparts. Feature importance obtained from nonlinear classifiers (e.g., gradient-boosting trees) strongly correlates with the causal strength of variables without differentiating cause and effect variables. Correlated variables can be causal for HF, but they are rarely identified as effect variables. These results can be used to add the causal explanation of variables important for ML-based prediction modeling.

We study the recovery of multiple high-dimensional signals from two noisy, correlated modalities: a spiked matrix and a spiked tensor sharing a common low-rank structure. This setting generalizes classical spiked matrix and tensor models, unveiling intricate interactions between inference channels and surprising algorithmic behaviors. Notably, while the spiked tensor model is typically intractable at low signal-to-noise ratios, its correlation with the matrix enables efficient recovery via Bayesian Approximate Message Passing, inducing staircase-like phase transitions reminiscent of neural network phenomena. In contrast, empirical risk minimization for joint learning fails: the tensor component obstructs effective matrix recovery, and joint optimization significantly degrades performance, highlighting the limitations of naive multi-modal learning. We show that a simple Sequential Curriculum Learning strategy-first recovering the matrix, then leveraging it to guide tensor recovery-resolves this bottleneck and achieves optimal weak recovery thresholds. This strategy, implementable with spectral methods, emphasizes the critical role of structural correlation and learning order in multi-modal high-dimensional inference.

IDS aims to protect computer networks from security threats by detecting, notifying, and taking appropriate action to prevent illegal access and protect confidential information. As the globe becomes increasingly dependent on technology and automated processes, ensuring secured systems, applications, and networks has become one of the most significant problems of this era. The global web and digital technology have significantly accelerated the evolution of the modern world, necessitating the use of telecommunications and data transfer platforms. Researchers are enhancing the effectiveness of IDS by incorporating popular datasets into machine learning algorithms. IDS, equipped with machine learning classifiers, enhances security attack detection accuracy by identifying normal or abnormal network traffic. This paper explores the methods of capturing and reviewing intrusion detection systems (IDS) and evaluates the challenges existing datasets face. A deluge of research on machine learning (ML) and deep learning (DL) architecture-based intrusion detection techniques has been conducted in the past ten years on various cybersecurity datasets, including KDDCUP'99, NSL-KDD, UNSW-NB15, CICIDS-2017, and CSE-CIC-IDS2018. We conducted a literature review and presented an in-depth analysis of various intrusion detection methods that use SVM, KNN, DT, LR, NB, RF, XGBOOST, Adaboost, and ANN. We provide an overview of each technique, explaining the role of the classifiers and algorithms used. A detailed tabular analysis highlights the datasets used, classifiers employed, attacks detected, evaluation metrics, and conclusions drawn. This article offers a thorough review for future IDS research.

Single-cell RNA sequencing (scRNA-seq) has revolutionized our understanding of cellular processes by enabling gene expression analysis at the individual cell level. Clustering allows for the identification of cell types and the further discovery of intrinsic patterns in single-cell data. However, the high dimensionality and sparsity of scRNA-seq data continue to challenge existing clustering models. In this paper, we introduce JojoSCL, a novel self-supervised contrastive learning framework for scRNA-seq clustering. By incorporating a shrinkage estimator based on hierarchical Bayesian estimation, which adjusts gene expression estimates towards more reliable cluster centroids to reduce intra-cluster dispersion, and optimized using Stein's Unbiased Risk Estimate (SURE), JojoSCL refines both instance-level and cluster-level contrastive learning. Experiments on ten scRNA-seq datasets substantiate that JojoSCL consistently outperforms prevalent clustering methods, with further validation of its practicality through robustness analysis and ablation studies. JojoSCL's code is available at: https://github.com/ziwenwang28/JojoSCL.

Educators teaching entry-level university engineering modules face the challenge of identifying which topics students find most difficult and how to support diverse student needs effectively. This study demonstrates a rigorous yet interpretable statistical approach -- hierarchical Bayesian modeling -- that leverages detailed student response data to quantify both skill difficulty and individual student abilities. Using a large-scale dataset from an undergraduate Statics course, we identified clear patterns of skill mastery and uncovered distinct student subgroups based on their learning trajectories. Our analysis reveals that certain concepts consistently present challenges, requiring targeted instructional support, while others are readily mastered and may benefit from enrichment activities. Importantly, the hierarchical Bayesian method provides educators with intuitive, reliable metrics without sacrificing predictive accuracy. This approach allows for data-informed decisions, enabling personalized teaching strategies to improve student engagement and success. By combining robust statistical methods with clear interpretability, this study equips educators with actionable insights to better support diverse learner populations.

Reinforcement Learning (RL) has demonstrated tremendous empirical success across numerous challenging domains. However, we lack a strong theoretical understanding of the statistical complexity of RL in environments with large state spaces, where function approximation is required for sample-efficient learning. This thesis addresses this gap by rigorously examining the statistical complexity of RL with function approximation from a learning theoretic perspective. Departing from a long history of prior work, we consider the weakest form of function approximation, called agnostic policy learning, in which the learner seeks to find the best policy in a given class $Π$, with no guarantee that $Π$ contains an optimal policy for the underlying task. We systematically explore agnostic policy learning along three key axes: environment access -- how a learner collects data from the environment; coverage conditions -- intrinsic properties of the underlying MDP measuring the expansiveness of state-occupancy measures for policies in the class $Π$, and representational conditions -- structural assumptions on the class $Π$ itself. Within this comprehensive framework, we (1) design new learning algorithms with theoretical guarantees and (2) characterize fundamental performance bounds of any algorithm. Our results reveal significant statistical separations that highlight the power and limitations of agnostic policy learning.

Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to neural dynamical systems that encompass modern large-scale diffusion and flow matching models. Despite the scalability of training, the generation of high-quality samples and their corresponding likelihood under the model requires expensive numerical simulation -- inhibiting adoption in numerous scientific applications such as equilibrium sampling of molecular systems. In this paper, we revisit classical normalizing flows as one-step generative models with exact likelihoods and propose a novel, scalable training objective that does not require computing the expensive change of variable formula used in conventional maximum likelihood training. We propose Forward-Only Regression Training (FORT), a simple $\ell_2$-regression objective that maps prior samples under our flow to specifically chosen targets. We demonstrate that FORT supports a wide class of targets, such as optimal transport targets and targets from pre-trained continuous-time normalizing flows (CNF). We further demonstrate that by using CNF targets, our one-step flows allow for larger-scale training that exceeds the performance and stability of maximum likelihood training, while unlocking a broader class of architectures that were previously challenging to train. Empirically, we elucidate that our trained flows can perform equilibrium conformation sampling in Cartesian coordinates of alanine dipeptide, alanine tripeptide, and alanine tetrapeptide.

Alignment via reinforcement learning from human feedback (RLHF) has become the dominant paradigm for controlling the quality of outputs from large language models (LLMs). However, when viewed as `loss + regularization,' the standard RLHF objective lacks theoretical justification and incentivizes degenerate, deterministic solutions, an issue that variants such as Direct Policy Optimization (DPO) also inherit. In this paper, we rethink alignment by framing it as \emph{distribution learning} from pairwise preference feedback by explicitly modeling how information about the target language model bleeds through the preference data. This explicit modeling leads us to propose three principled learning objectives: preference maximum likelihood estimation, preference distillation, and reverse KL minimization. We theoretically show that all three approaches enjoy strong non-asymptotic $O(1/n)$ convergence to the target language model, naturally avoiding degeneracy and reward overfitting. Finally, we empirically demonstrate that our distribution learning framework, especially preference distillation, consistently outperforms or matches the performances of RLHF and DPO across various tasks and models.