This chapter argues that the reliability of agentic and generative AI is chiefly an architectural property. We define agentic systems as goal-directed, tool-using decision makers operating in closed loops, and show how reliability emerges from principled componentisation (goal manager, planner, tool-router, executor, memory, verifiers, safety monitor, telemetry), disciplined interfaces (schema-constrained, validated, least-privilege tool calls), and explicit control and assurance loops. Building on classical foundations, we propose a practical taxonomy-tool-using agents, memory-augmented agents, planning and self-improvement agents, multi-agent systems, and embodied or web agents - and analyse how each pattern reshapes the reliability envelope and failure modes. We distil design guidance on typed schemas, idempotency, permissioning, transactional semantics, memory provenance and hygiene, runtime governance (budgets, termination conditions), and simulate-before-actuate safeguards.

This thesis examines self-attention training through the lens of Optimal Transport (OT) and develops an OT-based alternative for tabular classification. The study tracks intermediate projections of the self-attention layer during training and evaluates their evolution using discrete OT metrics, including Wasserstein distance, Monge gap, optimality, and efficiency. Experiments are conducted on classification tasks with two and three classes, as well as on a biomedical dataset. Results indicate that the final self-attention mapping often approximates the OT optimal coupling, yet the training trajectory remains inefficient. Pretraining the MLP section on synthetic data partially improves convergence but is sensitive to their initialization. To address these limitations, an OT-based algorithm is introduced: it generates class-specific dummy Gaussian distributions, computes an OT alignment with the data, and trains an MLP to generalize this mapping. The method achieves accuracy comparable to Transformers while reducing computational cost and scaling more efficiently under standardized inputs, though its performance depends on careful dummy-geometry design. All experiments and implementations are conducted in R.

Modern work relies on an assortment of digital collaboration tools, yet routine processes continue to suffer from human error and delay. To address this gap, this dissertation extends TheAgentCompany with a finance-focused environment and investigates whether a general purpose LLM agent can complete representative wealth-management tasks both accurately and economically. This study introduces synthetic domain data, enriches colleague simulations, and prototypes an automatic task-generation pipeline. The study aims to create and assess an evaluation set that can meaningfully measure an agent's fitness for assistant-level wealth management work. We construct a benchmark of 12 task-pairs for wealth management assistants spanning retrieval, analysis, and synthesis/communication, with explicit acceptance criteria and deterministic graders. We seeded a set of new finance-specific data and introduced a high vs. low-autonomy variant of every task. The paper concluded that agents are limited less by mathematical reasoning and more so by end-to-end workflow reliability, and meaningfully affected by autonomy level, and that incorrect evaluation of models have hindered benchmarking.

The goal of this thesis is to improve our understanding of the internal mechanisms by which deep artificial neural networks create meaningful representations and are able to generalize. We focus on the challenge of characterizing the semantic content of the hidden representations with unsupervised learning tools, partially developed by us and described in this thesis, which allow harnessing the low-dimensional structure of the data. Chapter 2. introduces Gride, a method that allows estimating the intrinsic dimension of the data as an explicit function of the scale without performing any decimation of the data set. Our approach is based on rigorous distributional results that enable the quantification of uncertainty of the estimates. Moreover, our method is simple and computationally efficient since it relies only on the distances among nearest data points. In Chapter 3, we study the evolution of the probability density across the hidden layers in some state-of-the-art deep neural networks. We find that the initial layers generate a unimodal probability density getting rid of any structure irrelevant to classification. In subsequent layers, density peaks arise in a hierarchical fashion that mirrors the semantic hierarchy of the concepts. This process leaves a footprint in the probability density of the output layer, where the topography of the peaks allows reconstructing the semantic relationships of the categories. In Chapter 4, we study the problem of generalization in deep neural networks: adding parameters to a network that interpolates its training data will typically improve its generalization performance, at odds with the classical bias-variance trade-off. We show that wide neural networks learn redundant representations instead of overfitting to spurious correlation and that redundant neurons appear only if the network is regularized and the training error is zero.

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These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads us to the framework of well-posedness and first considerations regarding reconstruction and inversion approaches. The second chapter then first deals with classical regularization theory of inverse problems in Hilbert spaces. After introducing the pseudo-inverse, we review the concept of convergent regularization. Within this chapter we then proceed to ask the question of how to realize practical reconstruction algorithms. Here, we mainly focus on Tikhonov and sparsity promoting regularization in finite dimensional spaces. In the third chapter, we dive into modern deep-learning methods, which allow solving inverse problems in a data-dependent approach. The intersection between inverse problems and machine learning is a rapidly growing field and our exposition here restricts itself to a very limited selection of topics. Among them are learned regularization, fully-learned Bayesian estimation, post-processing strategies and plug-n-play methods.

We introduce DAMCC (Deep Autoregressive Model for Dynamic Combinatorial Complexes), the first deep learning model designed to generate dynamic combinatorial complexes (CCs). Unlike traditional graph-based models, CCs capture higher-order interactions, making them ideal for representing social networks, biological systems, and evolving infrastructures. While existing models primarily focus on static graphs, DAMCC addresses the challenge of modeling temporal dynamics and higher-order structures in dynamic networks. DAMCC employs an autoregressive framework to predict the evolution of CCs over time. Through comprehensive experiments on real-world and synthetic datasets, we demonstrate its ability to capture both temporal and higher-order dependencies. As the first model of its kind, DAMCC lays the foundation for future advancements in dynamic combinatorial complex modeling, with opportunities for improved scalability and efficiency on larger networks.

This work presents a custom approach to developing a domain specific knowledge assistant for sustainability reporting using the International Financial Reporting Standards (IFRS). In this domain, there is no publicly available question-answer dataset, which has impeded the development of a high-quality chatbot to support companies with IFRS reporting. The two key contributions of this project therefore are: (1) A high-quality synthetic question-answer (QA) dataset based on IFRS sustainability standards, created using a novel generation and evaluation pipeline leveraging Large Language Models (LLMs). This comprises 1,063 diverse QA pairs that address a wide spectrum of potential user queries in sustainability reporting. Various LLM-based techniques are employed to create the dataset, including chain-of-thought reasoning and few-shot prompting. A custom evaluation framework is developed to assess question and answer quality across multiple dimensions, including faithfulness, relevance, and domain specificity. The dataset averages a score range of 8.16 out of 10 on these metrics. (2) Two architectures for question-answering in the sustainability reporting domain - a RAG pipeline and a fully LLM-based pipeline. The architectures are developed by experimenting, fine-tuning, and training on the QA dataset. The final pipelines feature an LLM fine-tuned on domain specific data and an industry classification component to improve the handling of complex queries. The RAG architecture achieves an accuracy of 85.32% on single-industry and 72.15% on cross-industry multiple-choice questions, outperforming the baseline approach by 4.67 and 19.21 percentage points, respectively. The LLM-based pipeline achieves an accuracy of 93.45% on single-industry and 80.30% on cross-industry multiple-choice questions, an improvement of 12.80 and 27.36 percentage points over the baseline, respectively.

These are the lecture notes that accompanied the course of the same name that I taught at the Eindhoven University of Technology from 2021 to 2023. The course is intended as an introduction to neural networks for mathematics students at the graduate level and aims to make mathematics students interested in further researching neural networks. It consists of two parts: first a general introduction to deep learning that focuses on introducing the field in a formal mathematical way. The second part provides an introduction to the theory of Lie groups and homogeneous spaces and how it can be applied to design neural networks with desirable geometric equivariances. The lecture notes were made to be as self-contained as possible so as to accessible for any student with a moderate mathematics background. The course also included coding tutorials and assignments in the form of a set of Jupyter notebooks that are publicly available at https://gitlab.com/bsmetsjr/mathematics_of_neural_networks.

In our times, when the world is increasingly becoming more dependent on software programs, writing bug-free, correct programs is crucial. Program verification based on formal methods can guarantee this by detecting run-time errors in safety-critical systems to avoid possible adverse impacts on human life and save time and money. This project work tries to leverage Abstract Interpretation techniques for static analysis of C programs. C Analyzer is a tool developed for static analysis of C programs. This implementation of C Analyzer provides a plug-and-play domain architecture for multiple abstract domains to be used. C Analyzer supports four abstract domains - Interval, Octagon, Polyhedra, and Bit Vector. We use these different domains for required precision in program verification. C Analyzer tool uses LLVM C/C++ compiler frontend Clang API to generate and traverse the Control Flow Graph (CFG) of a given C program. This tool generates invariants in different abstract domains for statements in basic blocks of CFG during CFG traversal. Using these invariants, some properties of a program, such as dividing by zero, modulus zero, arithmetic overflow, etc., can be analyzed. We also use a source-to-source transformation tool, CIL (Common Intermediate language), to transform some C constructs into simpler constructs, such as transforming logical operators, switch statements, and conditional operators into if-else ladders and transforming do-while and for loops into while loops. Using C Analyzer, C program constructs such as declarations, assignments, binary operations (arithmetic, relational, bitwise shift, etc.), conditions (if-else), loops (while, do while, for loop), nested conditions, and nested loops can be analyzed. Currently, this tool does not support arrays, structures, unions, pointers, or function calls.