Operator learning promises to revolutionize scientific computing by learning solution operators for differential equations directly from data, potentially accelerating tasks like design optimization and uncertainty quantification by orders of magnitude. The deep operator network (DeepONet), the first practical architecture for operator learning, consists of a trunk and branch network. Its output is given by a linear combination of basis functions, where the functions are learned by the trunk network and the coefficients are learned by the branch network. However, despite their theoretical promise, DeepONets suffer from poor accuracy compared to classical numerical solvers, limiting their practical adoption. Understanding and addressing these accuracy limitations is crucial for advancing the field.

In this thesis, we first analyze the performance limitations of the classical DeepONet. We demonstrate that for many classical examples, the trunk network's error is much smaller than the total approximation error. Thus, the space spanned by the basis functions contains functions which approximate the true solutions well. The total approximation error is dominated by the branch network's error, i.e., the error of the coefficients.

To investigate this further, we construct a modified DeepONet. In this modification we replace the learnable trunk network with optimal basis vectors (modes) derived from a singular value decomposition (SVD). This modification is called SVD-based operator network (SVDONet). This simplification enables us to decompose the total error into mode-specific contributions, revealing how the coefficients of different spatial modes are approximated. Our mode decomposition analysis yields several key insights.

First, we discover that for some modes a low training error does not necessarily correspond to a low test error, i.e., the coefficients learned for these modes do not generalize well.
Second, we show that architectural choices profoundly impact generalization: the standard "unstacked" DeepONet architecture, where all modes share hidden neurons, significantly improves generalization for modes corresponding to small singular values at the cost of the modes with large singular values.
Third, we study how improving the coefficients of one mode impacts the coefficients of other modes. Here, we show fundamental differences between different optimization algorithms.

These findings establish mode decomposition as a powerful lens for analyzing neural operators, revealing that the success of operator learning hinges not on learning all modes equally well, but on the delicate balance between mode prioritization, architectural coupling, and optimization dynamics.

Evolution can be modelled by semi-directed phylogenetic networks, partially directed graphs where directed edges represent reticulate evolutionary events. In this thesis, we present a polynomial-time algorithm that can reconstruct a level-2 semi-directed phylogenetic network from its quarnets (4-leaf subnetworks). First, we find a canonical form of a network by reducing our information to merely quarnet-splits. Here, we use the blob-tree of the network found with quarnet-splits to construct the canonical form of the network. Finally, we consider quartets, which give more information about level-1 and level-2 quarnets, to refine on the canonical form.

Accurate flood prediction is essential for effective risk management, but simulating with physics-based models is computationally intensive and time-consuming, limiting their use in operational use cases. To address this challenge, this study develops a deep learning surrogate model that replicates spatial flood depth predictions of the physics-based model SFINCS, with significantly reduced computational cost.

The surrogate model is trained with a synthetic dataset of 13,000 tracks generated by TCWiSE, that have been computed with SFINCS. Input for the surrogate model are key drivers of compound coastal flooding, including cumulative precipitation, maximum wind speed, and water depth at the river outlet, which are given to the model as scalar values. The results demonstrate that the model can reproduce flood depth patterns with a RMSE of 0.054 m on the original dataset and 0.069 m on a case study application. The CSI values of 0.829 and 0.776, respectively, are comparable to values reported in recent literature. Visual inspection of spatial predictions shows that the model performs well overall but shows localized error patterns, particularly in the western part of the domain, indicating that additional input features may be needed to better capture interactions in physical processes that occur in compound coastal flooding. From a computational perspective, the deep learning model runs approximately 100 times faster than the physics-based model SFINCS on CPU, with further gains on GPU. This computational efficiency allows for running more simulations within a given timeframe, enabling the exploration of an increased number of potential flood scenarios and providing decision-makers with more time to evaluate and respond.

Overall, the study demonstrates that deep learning surrogate models offer a promising alternative to physics-based models by providing accurate and faster predictions. Although the current set-up shows accurate results, there remains room for improvement through enhanced input representation, such as incorporating spatiotemporal information of the drivers or including additional hydrodynamic features. Further gains in performance and generalizability can also be achieved by refining the model architecture and training strategies.

Reconstructing high-resolution wind fields from sparse, low-resolution observations is a critical yet ill-posed problem in meteorological modeling. Classical approaches, such as Computational Fluid Dynamics (CFD), are often too computationally intensive to meet the demands of real- time or large-scale industrial applications. Meanwhile, conventional data-driven methods like Convolutional Neural Networks (CNNs) tend to produce overly smoothed outputs and struggle to recover fine-scale structures, especially under severe data sparsity.
This thesis explores the use of diffusion-based generative models for super-resolution in wind field reconstruction. A progressive SR3 (Super-Resolution via Repeated Refinement) frame- work is developed, combining a multi-stage architecture with stochastic denoising processes to gradually reconstruct high-resolution outputs. Extensive experiments demonstrate that the progressive SR3 consistently outperforms CNN-based baselines in terms of reconstruction accur- acy, perceptual quality, and robustness. Furthermore, a joint training strategy improves both performance and computational efficiency by enabling end-to-end optimization across stages.
The findings support the use of probabilistic diffusion models for meteorological super-resolution tasks and emphasize the effectiveness of progressive refinement in handling large upscaling factors. This approach provides a promising pathway for enhancing data-driven post-processing in atmospheric modeling.

A PIN pad is a common way to authenticate, in particular for mobile applications. To strengthen PIN authentication we utilize behavioural biometrics in the form of keystroke dynamics. For authentication we require new PIN entries from the actual user to be accepted, while entries from an adversary to be rejected. Hence we strive to capture a user profile as compact as possible, such that false acceptance rates (FAR) are low, while the true acceptance rate (TAR) remains high. We estimate a user profile P by training one-class classification methods on user data U. We compare three one-class classification methods, namely Multivariate Gaussian Estimation (MGE), Support Vector Data Description (SVDD) and k-Minimal Enclosing Ball (k-MEB). We collected PIN data and analysed nine users which entered their PIN correctly at least 50 times. We find all methods outperform dummy classifiers. In most cases, a TAR score of over 0.8 combined with a maximum FAR score of at most 0.1 is achieved after tuning. Therefore, we conclude adding a behavioural biometric check does increase security of PIN authentication substantially.

Accurately characterizing the properties of semiconductors at atomic resolution is crucial for advancing semiconductor technology. One of the key challenges in quantitatively interpreting Scanning Tunneling Spectroscopy (STS) is the influence of tipinduced band bending (TIBB) during semiconductor measurements. This thesis presents a numerical method that self-consistently solves the one-dimensional Poisson equation to correct for TIBB in adatom-covered semiconductors. The model uses the Block-SOR-Newton method to solve the nonlinear system of equations that arises from the discretization of the Poisson equation. Simulations demonstrate that the model can accurately describe the effect of the STM tip voltage on band bending. However, numerical instabilities were observed for high doping concentrations and surface state energies close to the Fermi level, attributed to overshooting of the Newton method. Potential solutions, such as using a Newton-Krylov method and adaptive grid refinement, were proposed to address these instabilities. Future work includes extending the model to non-equilibrium situations by introducing the full set of semiconductor equations and expanding to three dimensions to account for the STM tip geometry. These advancements would provide a useful tool for correcting STS data, ultimately deepening our understanding of semiconductor physics at the atomic scale.

This research aims to optimize a VRPTW that incorporates the driver satisfaction factors ‘region consistency’ and ‘workload preference’ while not increasing routing costs too much. The developed measures were optimized for using the ‘Random Allocation’, ‘Driver Assignment’ and ‘Integrated Approach’ methods for various weightings in a multi-objective setting. Driver satisfaction was explicitly optimized in the state-of-the-art VRPTW solver called PyVRP developed by ORTEC. The integrated approach outperformed both the driver assignment and random allocation methods on small and medium sized instances, whereas Driver Assignment performed best on large instances.

This thesis investigates the problem of time-varying function optimization. In particular, we study techniques to minimize the cumulative regret when optimizing a time-varying function in the Gaussian process setting. First, we introduce the problem and present a literature review of the current methods and results. Following this, we we propose enhancements to existing algorithms, demonstrating improved regret bounds. We discuss the applications of these algorithms and where they can provide a benefit compared to existing methods. With these applications in mind we introduce two new temporal models for time-varying functions and their associated algorithms. We test their performance in order to validate their effectiveness and potential benefits.

This thesis is on the subject of phylogenetic networks. These are schematic
visualisations used mainly to investigate the evolutionary history of species,
but which can be used for any set of distinguishable elements which have diverged from a common ancestor through some evolutionary process. The research specifically focuses on a way to encode these phylogenetic networks, called μ-representation, which enables researchers to efficiently compare networks in polynomial time. The main contribution of this thesis lies in demonstrating that there are certain classes of phylogenetic networks for which the μ-representation or a modified version thereof serves as a unique encoding and can therefore be used to generate a metric for comparison. Additionally, it is shown that these results do not extend to some other classes of networks. Furthermore, this research shows that certain other information can be gained from analysing the μ-representation of a network, such as which nodes are adjacent to so-called bridges or cut-edges, and what the in-degrees of the nodes in the network are.

In this thesis, we study a particle method for Bayesian deep learning. In particular, we look at the estimation of the parameters of an ensemble of Bayesian neural networks by means of this particle method, called Stein variational gradient descent (SVGD). This method iteratively updates a collection of parameters and it has the property that its update directions are chosen such that they optimally decrease the Kullback-Leibler divergence. We also study gradient flows of probability measures and show how gradient flows corresponding to functionals on the space of probability measures can induce particle flows. We formulate SVGD as a method in this space. In the regime of infinite particles we show results about convergence of SVGD. An existing convergence result for SVGD can be extended by showing that the probability measures, governing the collection of SVGD particles, are uniformly tight. We give conditions under which this holds.

To meet global green energy targets, the bottom founded offshore wind industry is looking for ways to economically expand markets to deeper waters. A reduction of the hydrodynamic load is necessary to achieve this. One option is to perforate the monopile around the splash zone. Here the related work of Q. Star is continued through the implementation of a 2D Navier-Stokes based LES CFD model with VMS closure in Gridap.jl. The numerical simulations are gathered to create a dataset on which a machine learning surrogate model is trained. The analysis does not include secondary design aspects such as manufacturing, installation, noise mitigation, scour, corrosion, acidification, or marine growth. Neither does it require secondary fluid phenomena such as 3D simulations, FSI, VIV, and fatigue.
Analysis of 2D perforated cylinders, built up parametrically using the wall thickness, angle of attack, diameter, inflow velocity, number of perforations and porosity as input parameters shows that the first two are of negligible influence, and number three and four can, at least for their mean values, be modelled accurately using Morison equations. A non-scaled diameter does prove important in reducing the random nature of frequency-based effects. The number of perforations and the porosity provide complex interactions both from a design-force mean and variability perspective, as well as for the frequencies and vibrations they generate. A 2D LES-VMS model gives the perfect trade-off between cost and accuracy, predicting a possible drag reduction of more than 50% compared to a closed cylinder with large diameter.
Different machine learning surrogate models are analysed with the goal of massively speeding up the analysis in the future. This is achieved by a factor of 350 thousand using Random Forests, Gaussian Processes and Neural Networks. Although the Gaussian Processes preliminary show the best accuracy, below 6% error for millisecond predictions, further work on Neural Networks could give them the advantage in future analyses.

The real-life Vehicle Routing Problem (VRP) is the problem in which a set of vehicles needs to perform a set of tasks such that we have a shortest total driving distance. Such problems can be solved using construction algorithms. Finding the best-performing construction algorithm is time-consuming because these algorithms consist of many different elements, which differ between algorithms. In this thesis, a Neural Network (NN) model is built that predicts the best-performing algorithm from a pre-determined set of algorithms for a specific input case. These predictions are based on problem features extracted from real-life data. For our first NN model, we evaluate the best settings with a grid search, which leads to a model with a mean-squared error of 0.147 and an accuracy of $58.6\%$. We try to improve this original model by data balancing and varying the input features. First, we balance the data using downsampling and oversampling performed by an Integer Linear Program (ILP), which does not lead to a better-performing NN. Secondly, we add more input features to the model, which leads to a slight improvement because the model has more information about the problem at hand. After, we perform an elaborate feature analysis using permutation feature importance, SHAP, and Greenwell numbers. Based on this analysis, we reduce the number of input features to only 12. This reduction leads to the best-performing model with a mean-squared error loss of 0.125 and an accuracy of 61.7\%. To investigate whether our prediction model indeed improves the routes using the predicted algorithm, we look at the distribution of predictions. For each company, we replace the algorithm in use with the algorithm most often predicted by our model. This replacement indeed improves the results for most of the considered companies.