Plastic waste transported to oceans through canals and rivers becomes increasingly challenging to retrieve and harmful to ecosystems. Catching systems designed by Noria Sustainable Innovators can be used to capture the plastics as close to their source as possible. Deciding the best locations to place these systems is a difficult task, which is why a model for location optimization of catching systems for plastic waste removal from waterways is designed in this thesis: the Plastic Waste Flow Capturing Location Model (PW-FCLM).

In this model, the plastic waste flow through a network of waterways is represented as a Markov chain, using environmental data as inputs to estimate the initial probabilities and transition probabilities of plastic waste in the network. The PW-FCLM extends an existing Markov Decision Process-based Flow Capturing Location Model by incorporating various types of catching systems, considering sensitive areas, and specifying orientations for the systems. The equivalence of the linearized version of the extended model is demonstrated, a proof of NP-hardness of the problem is given and a greedy heuristic is presented as an alternative solution method.

A sensitivity analysis on the different types of input parameters is performed, and the runtimes for different problem sizes and solution methods are tested for case studies of Delft and Groningen in the Netherlands. The model is most sensitive to changes in the distance between the nodes in the network and to the probability of getting stuck due to water vegetation. For budgets up to B=2 and problem sizes up to n=375 nodes, the exact optimal solution can be found efficiently without a commercial solver license. For larger problem sizes or a higher budget, the heuristic appears to be a more appropriate solution method.

For future research, it is recommended to further study the influence of the distance between the nodes on the optimal solution and to investigate the plastic flow representation in the Markov chain further. Exploring the model's application to larger areas, such as provinces or countries, would be beneficial. The real-life effectiveness of placing catching systems at the optimal locations suggested by the model, depends on the accuracy of the input parameters. In this thesis, the values of the input parameters are primarily based on estimations of experts. It would be beneficial for the user of the model to further validate the values of the input parameters through experiments.

In this thesis we implement the Lindblad equation in the matrix product state (MPS) formalism using an operator splitting method. We developed a second-order method based on a Trotter approximation and a third-order high-dimensional midpoint method and we proposed a new fourth-order method based on Duhamel’s principle and a nested RK4 method, all of which preserve positivity and Hermiticity of the density operator. We simulated spin transport through an XXZ-Hamiltonian Heisenberg chain, for which we found the magnetisation profile and measured a spin current of 0.04-0.05. The results obtained are consistent with the existing literature. The extensive error analysis shows that the time step ∆t is the main contributor to the error, if the bond dimension χ is set to at least 15. The third-order method is in general preferred to the second-order method, as only this method preserves trace. We also analysed the Hubbard model, including a spin orbit coupling, in order to propose a method for simulating the chiral induced spin selectivity (CISS) effect.

Community detection and graph partitioning have seamlessly integrated themselves into the fabric of network science by providing valuable insights into the structure, function, and dynamics of complex networks. In this thesis, a comprehensive performance comparison of the recently introduced Linear Clustering Process (LCP) is carried out against well-established clustering algorithms from literature. We evaluate its effectiveness using synthetic benchmarks commonly employed in the field, as well as real-world networks with both known and unknown community structures. Through our analysis, we reveal that the Linear Clustering Process consistently yields superior community partitions with optimized modularity when the clusters are well-defined compared to the majority of the assessed algorithms. Meanwhile, remarkably, this improved performance is achieved while maintaining computational complexity comparable to the simplest existing clustering algorithms. Furthermore, this thesis also provides an empirical approach for enhancing the performance of a variant of Linear Clustering Process on power-law networks.

This thesis aims to develop an advanced numerical solver capable of efficiently computing the resonant states of quantum mechanical two-body and three-body problems, thereby expanding our understanding of these complex systems. The quantum three-body problems feature at least two dimensions, which necessitates substantial computational efforts. Therefore, in order to tackle these challenging computations, we need to seek assistance from supercomputers. By harnessing the capabilities of high-performance computing, we can significantly reduce the amount of time spent waiting for programs to run for hours.

In this thesis, we first introduce some basic knowledge about quantum few-body problems and resonant states, showing how the physical problem gives rise to a mathematical problem, the quadratic eigenvalue problem (QEP). Building upon the physical background, our journey in developing the methodology begins with two fundamental components: discretization and eigensolver. The pseudo-spectral methods are introduced to represent the quadratic eigenvalue problem as a matrix problem, by which we can solve the problem numerically through some eigensolvers. We describe a classical approach called linearization for solving QEPs, which transforms the quadratic problem into a generalized eigenvalue problem. Following the linear transformation, we apply the Jacobi-Davidson QZ (JDQZ) method, an iterative eigensolver, to solve the linearized problem. Alternatively, we could also use the Jacobi-Davidson (JD) method to approximate the quadratic eigenvalue problem's eigenpairs directly. In this thesis, we provide an outline of the Jacobi-Davidson process for solving both linear and quadratic problems. Two routes for solving QEPs are utilized and compared: linearization combined with the Jacobi-Davidson QZ method, and the quadratic Jacobi-Davidson method. Through our research and analysis, we demonstrate that the Jacobi-Davidson algorithm exhibits superior computational efficiency when adapted to solve QEPs directly.

Another significant objective of this thesis is to parallelize the eigensolver on supercomputers. We implement a hybrid distributed/shared memory parallelization of the Jacobi-Davidson algorithm to solve quadratic eigenvalue problems that arise from one-dimensional three-body problems. We leverage the tensor structure inherent in the three-body problem to optimize computational efficiency. Specifically, we implement an efficient tensor product scheme for the application of the stiffness and damping operators, which are realized as dense matrix-matrix products. By incorporating a preconditioner that also preserves the tensor structure, we enhance the performance of our Jacobi-Davidson algorithm in computing three-body resonance poles within an acceptable speed.

Intensity modulated proton therapy is an advanced radiotherapy technique that is used to treat cancer patients. In order to successfully treat a patient, sufficient dose to the tumor is required. However, during the fractionated treatment, multiple errors can cause a difference between the planned and actual dose delivery. To ensure adequate dose delivery in potential error scenarios, robust treatment plans are acquired as these are less sensitive to uncertainties inherent to proton therapy. However, robust optimization is challenging.

First, as robust optimization accounts for multiple error scenarios, the time needed to generate optimal treatment plans increases significantly. Therefore, it is investigated in this thesis if the optimization time can be reduced while preserving treatment plan quality. This is investigated through two different methods. In the first method, the number of error scenarios accounted for during optimization is reduced. We found that this can significantly reduce optimization time, while improved target coverage and lower risk on side effects are obtained. However, the near-maximum dose to the tumor was found to be less favourable. The second method investigated is variance optimization. This method significantly reduces optimization time. However, for similar target coverage, the risk of side effects increases.

Another challenge related to robust optimization is the increase in delivered dose to healthy tissues surrounding the tumor, which increases the risk of side effects. Therefore, it is investigated in this thesis if the risk of side effects can be lowered by allowing higher maximum dose to the tumor. It is found that this method indeed reduces the risk of side effects. However, the increased maximum dose to the tumor may not be clinically desired as the increase may lead to higher risks of other side effects: edema and fibrosis. The clinically desired trade-off between near-maximum dose and normal tissue sparing should be established.

The controller placement problem concerns the placement of controllers on Software-Defined Networks such that a pre-defined objective is optimized. In this thesis, we conduct research on the controller placement problem with network availability as the performance metric. Unlike other approximate evaluations, we compute the exact value with the path decomposition algorithm, which allows us to accurately measure the quality of different placements. After that, we investigate on the graph metrics' effect on network availability and develop a placement strategy based on degree and distance. Greedy algorithm and genetic algorithm are also introduced to address the controller placement problem. We analyze the optimal placement of OS3E network and other 100 real-world networks. We find that different placements affect availability a lot, which indicates that it is necessary to find a strategy to place controller such that a near-optimal placement is achieved. Finally, four placement strategies are tested on Erdős–Rényi random graphs, Barabási–Albert random graphs, and 155 real-world graphs from the Topology Zoo. Results show that the performance of these four strategies is almost same for most networks. However, the complexity of these four methods is very different, which suggests that the controller placement strategy based on graph metrics is efficient.

The main goal of this thesis is to understand the topological properties of semantic networks, to find language-specific patterns, and to investigate their connection principles. Interpreting unstructured texts in natural language is a crucial task for computers. Natural Language Processing (NLP) applications rely on semantic networks for structured knowledge representation. Although NLP technologies have been applied to various domains with some degree of success, they still face many challenges due to the ambiguity of human language. To inform better algorithms, we need to pay attention to fundamental structures of semantic networks in different languages. However, these remain to be investigated properly. In this thesis we extract semantic networks with 7 distinct relations for 11 languages from ConceptNet. We systematically analyze the degree distribution, degree correlation and clustering of these networks. We also measure their structural similarity and complementarity coefficients. Our findings show that semantic networks have universalities in basic structures: they have high sparsity, high clustering, and power-law degree distributions. Our findings also show that the majority of the considered networks are scale-free. In addition, our results show that networks in different languages exhibit different properties, which are determined by grammatical rules. For example, the networks of highly inflected languages show peaks in the degree distributions that deviate from a power-law. Furthermore, we find that depending on the type of semantic relation and the language, the connection principles of networks are different. Some networks are more similarity-based, while others are more complementarity-based. We conclude the thesis by demonstrating how the knowledge of similarity and complementarity can better inform NLP in link prediction tasks.

In an attempt to find alternatives for solving partial differential equations (PDEs)
with traditional numerical methods, a new field has emerged which incorporates
the residual of a PDE into the loss function of an Artificial Neural Network. This
method is called Physics-Informed Neural Network (PINN). In this thesis, we study dense neural networks (DNNs), including codes developed in the context of this bachelor project. We derive the backpropagation equations necessary for training and use different configurations in a DNN to test its interpolating accuracy. We distinguish between a-PINNs which use automatic differentiation to evaluate a PDE, and n-PINNs which approximate differential operators in a PDE with numerical differentiation. We compare both PINNs on the harmonic oscillator, the 1D heat equation and the 1-soliton and 2-soliton solutions of the Korteweg-De Vries (KdV) equation. Both PINNs could accurately converge to the solution, except to the 2-soliton solution, where the a-PINN outperformed the n-PINN. Furthermore, we tested a highly nonlinear problem of the KdV equation, which can be described by a train of solitons. We observed that PINNs are inaccurate if insufficient training samples are used for training. Adding training samples on the interior from a numerical solution leads to a good qualitative agreement, though more effort is required to find a better network configuration to obtain more accurate predictions.
Additionally, PINNs were used for inverse problems to derive an unknown coefficient in a PDE and proved to be highly accurate for noiseless data. When we
generated training samples with 10% noise from a uniform distribution, the PINN
results’ relative error stayed within a margin of under 2%. However, inverse PINNs are much more inefficient compared to nonlinear least squares methods like the Levenberg–Marquardt algorithm.
As of now, PINNs are still very early in development and stand no match against
traditional numerical methods to a known PDE. They may, however, provide a
useful alternative in the future as they are constantly being improved.

For scaling up the qubits in silicon quantum computers, it is vital to determine crosstalk effects that can lower the fidelity of the computer.
In this computational project, we examine single-qubit gate-fidelities in the presence of crosstalk for uncoupled spin qubits that are driven with X-gates via electron dipole spin resonance (EDSR). We introduce two models: the first model introduces the AC Stark shift and the novel second model expands on this by adding a resonance frequency shift on top. We assume the latter resonance frequency shift to be due to heating effects. We optimize the gate-fidelity for a qubit coupled to up to six drives as a function of the overall driving time and -frequency of a single drive for both models using the Nelder-Mead algorithm.

Using the AC Stark shift model, we still obtain 0.99999 fidelity if we do not account for the crosstalk. However, when using the second model, the fidelity drops to 0.69 in the presence of two drives when we do not correct for the heating-induced resonance frequency shift and the AC Stark shift. Furthermore, the fidelity decreases linearly with the number of drives coupled to the qubit, implicating that the resonance frequency shift will become a significant problem for the scalability of silicon quantum computers. We find that we can correct for the resonance frequency shift entirely by using optimized driving time and -frequency, where most gain comes from optimizing the driving frequency. Moreover, we discover that there is a linearly increasing dependence of the resonance frequency shift at the theoretical driving time as a function of the total drives. Up to a translation factor of 0.5 MHz, we discover the same linear relationship for the correction needed on the theoretical driving frequency to hit maximum fidelity as a function of the total drives.

Chiral induced spin selectivity (CISS) is an unexplained phenomenon in physics, in which the interaction within chiral molecules brings about a selectivity in the spin of the electrons.
The Breit interaction has been suggested as a possible explanation, but has not yet been researched in the context of CISS.
In this thesis, first a derivation of the Breit equation is given, which in the second part is being used to construct a simulation of electron transport through a simple chiral molecule.
The spin-polarization for the transmission including the Breit interaction was compared to the case excluding the Breit interaction, and the two were found to be substantially different. From that it followed that the influence of the Breit interaction on the transport of an electron through a chiral molecule is significant and it should therefore not be disregarded in calculations on CISS.

The Curie-Weiss model is a simplification of the Ising model to show the existence of a phase transition for ferromagnetism. In this thesis, we study the behaviour of sums of these dependent variables. We prove in general that under the appropriate assumptions, we can still conclude a version of the Law of Large Numbers. We also find that if there exists a certain m∈R, λ>0 and integer k≥1, we have that (S_n-nm)/n^1/2k converges to exp(-λs^2k/(2k)!) in distribution.
For the Curie-Weiss model this means that for β, which is a constant proportion to inverse temperature, we find that if β∈(0,1) we have S_n/n→δ(s) and S_n/√n→ N(0,σ²) in distribution where σ²=(1-β)^-1-1. At β=1 there occurs a phase transition, we still have that S_n/n→δ(s), but now S_n/n^3/4→\exp(-s⁴/12). When β>1 we can find an m>0 such that S_n/n→½[δ(s-m)+δ(s+m)].
We also study the Curie-Weiss model where we assume that it is under the influence of a magnetic field. We prove that we do not find a phase transition, and we always have S_n/n→δ(s-m) in distribution for some m∈R. Next to this we find that (S_n-nm)/√n always converges to a normal distribution.

Current research on osteoporosis detection and transcranial imaging suggests that it is necessary to find a method to calculate the speed of sound in curved bones. A popular method to find the speed of sound in bones is by the bidirectional axial transmission technique. Currently, this method assumes that the arrival time of the head wave depends linearly on the source-receiver distance. For curved bones this assumption is not valid. The arrival time of the head wave as a function of the source-receiver distance follows a curved trajectory. In this report, we attempt to find an extension to the current method that also works for curved bones, and evaluate how accurate the results of this extended method are.

This extension is based on the semblance method, that quantifies how well a certain trajectory agrees with the received data. In the first part of our project, theoretical trajectories are calculated for flat and semicircular bones. Furthermore, a numerical method is discussed to find these trajectories for arbitrary geometries. Comparing the trajectories corresponding to different velocities using the semblance methods allows for the determination of the speed of sound.

The accuracy of this extended method is evaluated by computer simulations. In these simulations, a Philips P41 Cardiac Sector Probe consisting of 96 elements was used to perform measurements on three different geometries. These geometries consisted of three different single interfaces between water (simulating soft tissue) and bone. The first interface consists of a straight interface between a 2 mm thick layer of water and a 5 mm thick layer of bone (that continues until the bottom edge of the simulation). The second simulation is the same, but the interface is rotated 2 degrees around its midpoint. In the third and fourth simulation, the interface is a semicircle with a 50 mm and 40 mm radius, respectively. The speed of sound in the bone was taken to be 3 mm/s. The received data was interpolated by a cubic interpolation with a refinement of 4. Both the bidirectional and the extended,
trajectory-based method were applied on the raw and interpolated data.

For the flat interface, the bidirectional method performed slightly better than the trajectory-based method (0.13% and 0.30% error, respectively). The same is true for the tilted interface (0.16% and 0.23% error). Interpolation did not affect the results here. For the 50 mm semicircular interface, the trajectory-based method performed much better than the bidirectional method: it had a 0.10% error compared to a 2% one. This was also true when the interpolation was applied: then the errors were 0.3% and 1.97%. Similar
improvements were seen for the 40 mm radius semicircle. The higher error of the trajectory-based method for the flat and tilted interfaces can be explained by small errors in the additional assumptions that this method makes.

One drawback of the trajectory-based method is that there can be problems interpreting the results. This can be seen in the semblance plots, which plot the semblance coefficient (how well the trajectory fits the data) against the test velocity. For the bidirectional method, these plots clearly have a single biggest peak. For the trajectory-based methods the plot contains multiple large peaks. Furthermore, for the semicircular bone this plot contains a flat peak, which can cause multiple test velocities to have approximately the same semblance. This is a problem, since it means that small changes to the experimental setup can cause big changes in the calculated velocity. Indeed, this can be seen for the trajectory-based method on the 40 mm radius semicircular bone, which has a 1.6% error without interpolation (0.43% with interpolation). Further research could examine solutions to this problem, such as increasing the window size of the semblance method, taking the average velocity over flat peaks, or using an alternative to the semblance method. Finally, we conclude that the trajectory-based method represents an improvement over the bidirectional method when examining curved bones.

This thesis contains a rigorous derivation of the path integral formulation of the Isingmodel with multiple original proofs. Besides that, thesis also contains various resultsof simulations of the 2D square lattice Ising Model with nearest-neighbour interactionsusing the Swendsen-Wang algorithm. Using finite size scaling to find critical exponentγ, we reported a value of γ = 1.748 ± 0.004. After calculating the relaxation times τ forvarious thermodynamic variables, we found the value for the dynamic critical exponentz to be in the range of z = 0.180 ± 0.004 and z = 0.282 ± 0.005.

The behaviour of the dilute Curie-Weiss model has been analysed for Gaussian, bounded and subGaussian couplings. This model is an abstraction of the Curie-Weiss model which is a model of a spin configuration where instead of a constant coupling between spins the spin coupling is a realisation of a random variable. To analyse this behaviour first the behaviour of the standard Curie-Weiss model is depicted. A notable part from this behaviour is the presence of a phase transition., indicating that this model can be used for a study of phase transitions. After having analysed the behaviour of the standard Curie-Weiss model theorems, stating the closeness between the standard Curie-Weiss model and the randomly dilute Curie-Weiss model for Gaussian, bounded and sub-Gaussian couplings, will be proven. Indicating that the models behave approximately the same way. These theorems will be in the form of a bound for the on randomly dilute Curie-Weiss model as an exponential multiplied by the standard Curie-Weiss model with the probability of this bound being a sub-Gaussian distribution.

In this thesis, the semilinear Cahn-Hilliard-Gurtin equation is studied using the method of Maximal Regularity. In 2012, Wilke developed a linear theory in $L^p$-spaces, and achieved a local and global well-posedness result for large $p$. In 2013, Denk and Kaip developed a linear theory in mixed integrability $L^pL^q$-spaces, using the method of Newton polygons. In this thesis, we connected the recent weighted anisotropic Mikhlin multiplier theorem of Lorist with the method of Newton polygons, leading to a linear theory in time-weighted $L^pw_\alpha L^q$-spaces, which is novel. By a postulation that the linear theory also holds in domains, we are able to treat the local well-posedness in the recently developed critical space setting of Prüss et al. This approach draws upon recent advances in interpolation theory in the setting of fractional Sobolev spaces with power weights in time, such as exhibited in the work of Agresti and Veraar.By adapting the global well-posedness result of Wilke, we are able to treat the semilinear equation in less regular spaces, i.e. smaller integrability parameters $p$ and $q$, and with rough initial data.

In this thesis, we present a study to obtain a clear and accurate overview of the progress and behaviour of COVID-19 in the Netherlands. We distinguish two parts for this study. The first part is to estimate the total number of infected people as a function of time by combining data from hospital admissions, daily reported cases and serological data. Using these data sets, we found that our estimation for the number of infected people was comparable to the estimations provided by the RIVM and Sanquin. Furthermore, we found that on average only 39.3% of the total number of cases were detected. 1.2% of the total number of infected people is admitted to the hospital and 18.6% of the hospitalized patients is admitted to the ICU. The second part is to develop a representative model that reproduces the estimated total number of infections using a modified SEIR model. These modifications include modelling the infection rate β(t) as a function of time using a simple linear ODE, a system of ODEs inspired by the Lotka-Volterra equations, the implementation of gamma distributed exposed and infected stages and lastly the incorporation of spatial heterogeneity. We found that our Lotka-Volterra inspired model was able to model multiple consecutive waves, which differs from the standard compartmental models. The other modifications however seemed to have only minor effects on the model and had some difficulties with matching historical data. We conclude that our Lotka-Volterra inspired model should be used to model consecutive waves for a longer period of time. The other modifications can be used to optimize the model.

The porous medium equation $\dv{t}u=\dv{x}(k(u)\dv{x}u)$ is a non-linear degenerate parabolic partial differential equation. Consequently, existence and uniqueness of its solutions is not immediately evident.
This bachelor thesis presents a detailed discussion of Atkinson's and Peletier's 1971 article ``Similarity profiles of flows through porous media" on existence and uniqueness of self-similar solutions to the porous medium equation.
First, in chapter 2 the general version of the porous medium equation along with some applications will be discussed. Then, in chapter 3 the proofs and statements of the Picard-Lindelöf theorem, Peano's existence theorem and Gronwall's inequality will be presented. These standard theorems concern differential equations and will be used in the next chapter. Finally, in chapter 4 Atkinson's and Peletier's article will be worked out in detail.

The core-periphery structure is a mesoscale topological structure that refers to the presence of a dense core and a sparse periphery. The core-periphery structure has been discovered in financial, biological, and technological networks. Various methods for detecting core-periphery structure have been proposed, exploring different discrete or continuous models and single or multiple core-periphery groups. A method that implements multiple-group detection and edge-direction dependency to core-periphery detection is yet to be researched. This report proposes an algorithm to extract the core-periphery structure that satisfies multiple-group and edge-direction dependency requirements. This algorithm features a heuristic process and can process a large-scale network in an acceptable amount of time. Details of the theory behind the proposed algorithm are presented. The algorithm is tested on synthetic, random scale-free, and sampled dark web networks to verify the basic and advanced feasibility. Finally, in-depth analysis with the knowledge of core-periphery structure on a large-scale dark web network sample is presented.

Celmigratie wordt voornamelijk door twee eiwitten veroorzaakt, namelijk Rac en Rho van de Rho-GTPases familie. Deze eiwitten kunnen zowel in actieve als inactieve vorm voorkomen [2], [6], [7]. De actieve vorm is gebonden aan het celmembraan en de inactieve vorm zit los in de cel [7]. De eiwitten kunnen zich verplaatsen in de cel. Doordat de actieve vorm gebonden is en de inactieve vorm niet, is de diffusiesnelheid van de inactieve vorm van het eiwit hoger dan van de actieve vorm [6], [7]. De kant met de hoogste concentratie actieve Rac is de voorkant van de cel. Actieve Rac zorgt voor een uitstulping aan de voorkant van de cel en Rho zorgt voor samentrekking van de cel aan de achterkant [2], [7]. Cellen zijn omgeven door een heel netwerk aan eiwitvezels waarmee de cel continu in aanraking is tijdens de migratie. Dit netwerk aan eiwitvezels wordt ook wel het extracellulaire matrix (ECM) genoemd. Doordat actieve Rac de cel stimuleert om met een speciale receptor (integrin) een binding aan te gaan met bijvoorbeeld het eiwit fibronectine en Rho deze uiteindelijk weer verbreekt, kan de cel ’lopen’. Eiwitten Rac en Rho werken elkaar tegen en remmen ook elkaar . Het ECM stimuleert juist het eiwit Rho en soms ook Rac [2]. In een specifiek lab experiment konden melanoomcellen (huidkankercellen) drie verschillende fenotypes hebben, namelijk random, oscillerend en polair [2], [5], [7]. Het fenotype random betekent dat de cel nagenoeg stilstaat, oscillerend betekent dat de cel wisselt tussen naar voren en naar achter gaan, en bij polair gaat de cel blijvend één kant op. In hetzelfde lab experiment is er sprake van een gradiënt in de afstand tussen pilaren bedekt met fibronectine, wat zorgt voor meer contact van een cel met fibronectine [5]. Dit bootst de invloed van het ECM na [2]. Als de cel het fenotype polair heeft en de kant op gaat van het ECM waarbij de afstand van deze pilaren bedekt met fibronectine steeds groter wordt, spreken we van topotaxis [7]. Als de cel de andere kant hierbij op gaat, spreken we weer van tegengestelde topotaxis. Of er sprake is van topotaxis of van tegengestelde topotaxis hangt af van in hoeverre het ECM het eiwit Rac en/of Rho stimuleert. In experimenten is gevonden dat topotaxis een rol speelt in het verspreiden van kankercellen zoals melanoomcellen [4]. De soort bindingen die een cel met zijn integrinnen maakt met het fibronectine in het ECM kunnen worden gezien als ’catch-slipbindingen’. Catch-bindingen zijn bindingen die sterker worden als de kracht erop toeneemt tot een bepaald moment. Dan is de kracht zo groot dat de binding breekt. ’Slip’-bindingen daarentegen worden zwakker als de kracht toeneemt. Deze combinatie weergeeft volgen experimenten een goed beeld van de werkelijke binding tussen een integrin en fibronectine [3]. We modelleren aan de hand van een aantal verschillende wiskundige modellen celmigratie met behulp van de eiwitten Rac en Rho en het ECM. We doen dit voor een 2-compartimenten model zonder diffusie en alleen afhankelijk van de tijd, dat de koppeling tussen de voor- en achterkant van een cel beschrijft, en een model in 1D met diffusie. Deze modellen bestaan uit reactie-diffusie differentiaalvergelijkingen en we hebben ze met behulp van python 3.9 gemodelleerd. We onderzoeken de onderliggende verbanden tussen de parameters uit de modellen en experimentele bevindingen. Op deze manier willen we bereiken dat we het fenotype kunnen voorspellen en zo kanker vroeger kunnen constateren bij mensen en beter therapieën ertegen kunnen ontwikkelen.

In this work neural networks are used to approximate the solutions of multiple differential equa- tions. Three main differential equations are investigated: the mass-spring equation, the heat equation and the Navier-Stokes equations coupled with the continuity equation. Each of these equations are approached differently with the current methods, therefore these equations test the applicability of the newly proposed method on a general differential equation. The results of various experiments performed in this work show that the method is able to approximate to great precision the mass-spring equation as well as the heat equation. However, the method does not perform well on the Navier-Stokes equations coupled with the continuity equation. The main advantage of this method is that meshes used by this method can often be very small compared to other approximation algorithms. When a slight decrease in performance is allowed, the method can be made completely mesh free. A huge drawback of the method is that there is no guarantee that the method will generate a reasonable approximation, for a given neural network configuration. This makes it necessary to look for a good configuration of the neural network. At this moment, this can only be done using trial and error and is therefore very time-consuming. Sometimes no good configuration is found at all. Another drawback is that the generated approximation does not satisfy the boundary and initial conditions. Multiple ways are proposed in this work to tackle this problem, but they introduce much more complexity and a performance decrease. The neural networks used in this work are generated using software called PyTorch, which comes with an automatic analytic differentiation tool. This part of PyTorch makes it possible to approximate differential equations using neural networks.