In this thesis, a deep learning-based surrogate model for predicting sea ice dynamics is developed that is capable of predicting linear kinematic features in a high-resolution setting. Predicting sea ice dynamics at high resolutions is critical for understanding climate patterns and enabling safe navigation in Arctic regions. Traditional continuum models based on the viscous-plastic rheology are computationally intensive when employed at high resolutions to capture linear kinematic features (LKFs), which are narrow zones of deformation in sea ice.

A supervised learning approach was adopted, focusing on convolutional neural network architectures, specifically the U-Net and a classic bottleneck model. Various loss functions were explored, including traditional metrics like the MSE and novel domain-specific functions such as the strain rate error (SRE), which incorporates physical knowledge of sea ice behaviour. The models were trained on a dataset generated from numerical simulations of sea ice dynamics on a 2 km grid.

The key findings indicate that the U-Net architecture combined with the MSE+SRE loss function outperforms other models. This architecture's depth and use of skip connections allow it to capture complex, multi-scale patterns inherent in sea ice dynamics. The integration of the SRE into the loss function significantly enhances the model's ability to predict LKFs, demonstrating the benefit of incorporating domain knowledge into machine learning models.

While the surrogate model effectively predicts LKFs for short-term forecasts up to approximately 5.5 hours (10 time steps), errors accumulate over longer periods, leading to significant errors after about 11 hours (20 time steps). However, the efficiency gain of this surrogate model is striking: the computation of 10 time steps can be done within a second, compared to the 30 minutes that traditional numerical methods need. The study also underscores the critical importance of training data selection and preparation in influencing model performance and generalisation capabilities.

In conclusion, this work demonstrates that a deep learning-based surrogate model, particularly utilising the U-Net architecture and the MSE+SRE loss function, can effectively predict sea ice dynamics at high resolutions with significant computational efficiency. The model generates forecasts within seconds, offering a viable alternative to traditional numerical simulations that require hours of computation. Future work should focus on mitigating error accumulation to extend the forecasting horizon and exploring advanced learning methods to further integrate physical insights into the modelling process.

In phylogenetics, it is important to figure out what method to obtain a phylogenetic tree, a tree showing how species evolve from one another, is more reliable. A phylogenetic tree is a mathematical tree, where every internal vertex has at least 2 children, and the leaves are labeled bijectively with a set $X$. One useful tool to help determine what method is more reliable is to find the rooted triplet distance between two trees. If the distance from a newly developed tree is far from what we know to be reliable trees, then this method is probably unreliable. For this distance, one should look at how many triplets the trees do not have in common. In this report, we will look at a specific algorithm to compute the rooted triplet distance between two special rooted phylogenetic trees, caterpillars, by checking how many triplets these two caterpillars have in common. In this report, we will map the leaves into $\mathbb{N}^3$ and $\mathbb{N}^d$, to determine how many triplets 3 or $d$ trees have in common. Especially for $\mathbb{N}^d$ it becomes complicated to decide for what regions of $\mathbb{N}^d$ the leaves need to be counted, and for which ones not. After finding the closed-form notation for this, we will also discuss how many terms there are in the sum of this equation. Lastly, we will talk about the time complexity of this algorithm. The idea is that this algorithm will work faster than the naive approach, which runs in $O(n^3)$ time. Due to complications in the third and higher dimensions, this faster running time may or may not be accomplished, depending on whether another algorithm can be found to resolve these complications, which will not be done in this report. A closed-form notation can be found to express the number of triplets two caterpillars do not have in common, and the number of terms this equation has will also be discussed in Chapter 5.

The level-set (LS) method uses a signed-distance function to capture the interface in two-phase flows. Geometrical properties can be easily obtained, and merging and splitting of the interface are handled automatically by the LS method. However, it is not inherently volume conserving. Several methods found in literature that aim to solve this problem are discussed in this report, including the interface-correction level-set (ICLS) method. The ICLS method uses an additional advection step with a correction-velocity field to restore global volume loss/gain. We present the volume-of-fluid-based local interface-correction level-set (VOF-LICLS) method. This is an extension to the ICLS method that aims to restore volume locally. This is achieved by coupling the ICLS method with the VoF method. In each time step we evolve both the level-set function and the volume fraction function. The VoF advection is performed using a Lagrangian-Eulerian method on a dual mesh. From the advected LS field volume fractions are constructed and compared to the advected VoF field. This allows us to use local volume fluxes for the velocity field, instead of a global volume flux. The construction of the correction-velocity is performed with the use of an analytic equation and the local volume fluxes. The novel volume correction procedure is then executed iteratively. The level-set equation is discretized in space with a finite element approach. Instabilities occur for the standard Galerkin approach for pure advection equation, so the SUPG method is used in order to obtain stable solutions. The performance of the developed method in this thesis is compared with LS and ICLS for three different test cases. We report a significant improvement for local volume conservation, and we observe that VOF-LICLS yields more accurate interface positions.

Chemotaxis is een verschijnsel uit de biologie dat zich op verschillende wijze
voordoet in het menselijk lichaam en bij bacteriën. Er is onderzocht hoe het
ééndimensionaal geval op te lossen is met de discontinue Galerkinmethode. Hierin
is het model eerst behoorlijk versimpeld en steeds verder uitgebreid. Zo is gekeken
naar de discontinue Galerkinmethode en de toepassing op de transportvergelijking
en kleine aanpassingen hierop én op de toepassing op de warmte
vergelijking.