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.