Sensitivity analysis and uncertainty quantification of nuclear reactors requires many expensive high-fidelity simulations. To approximate the dynamics of such a time and parameter dependent system efficiently and effectively, reduced order modelling (ROM) is used. In previous research, a ROM was constructed which used a combination of proper orthogonal decomposition (POD) and a locally adaptive sampling strategy based on sparse grids. To improve the representation of the physics in local parts of the parameter domain, in this thesis, the previous ROM is altered to use multiple local bases instead of one fixed global basis covering the entire parameter domain.

The spatially dependent local bases and local time dependent coefficients are interpolated separately in the parameter domain using the method of interpolation on a tangent space to the Grassmann manifold (ITSGM). The separate interpolators of the local bases and local coefficients are coupled in a space-time coupled approach. The algorithm based on sparse grids is modified so that it can adaptively draw new tangent planes in the parameter domain in a hierarchical manner. This results in the domain being split up into into smaller overlapping subdomains, each having their own tangent plane, number of basis vectors, and interpolators that use radial basis functions (RBF). The algorithm was tested on the Burgers Equation and the Molenkamp test which have an analytical solution. Then, the algorithm was tested on a numerically solved 2D neutron diffusion problem. First the parametric dependence of the modes and coefficients was analyzed, after which the performance of the algorithm was evaluated on these models via different experiments.

The results of the experiments on the 1D Burgers equation and the 2D neutron diffusion problem showed that the algorithm can adaptively and hierarchically draw new tangent planes and can accurately interpolate to new unknown solutions in the subdomains in both a 1D and 2D parameter setting. Higher order non-linear modes and coefficients are harder to interpolate than lower order modes and coefficients. The performance of the interpolator also depends on a combination of the chosen RBF and the size of the subdomains. The results from the Molenkamp test showed that in the smooth setting the new algorithm achieved a higher error with a higher number of evaluations, while in the steep setting the new algorithm performed on par with the previous algorithm, only if the interpolation accuracy threshold is set lower. The results from the neutron
diffusion problem indicate that the first principal angle can act as an indicator of which parts of the parameter domain contain more relatable physics than other parts of the domain.

The dependence of RBF interpolator on the subdomain size is caused by how the RBF values scale for further distanced points from the point of interpolation interest. Also more higher order modes than necessary can be included in the local basis interpolation method without worsening the interpolation accuracy of lower order modes, given that no numerical noise is present in the local bases or coefficients. Interpolating the time-dependent behaviour can be easier in the
space-time coupled approach than in the approach based on the global basis. However, a lower interpolation accuracy threshold should be chosen as full time evolutions are being interpolated instead of single state vectors. The current interpolation method scales worse than the previous algorithm, as the corner points of the parameter domain will always have to be sampled when using the RBF interpolator compared to local linear basis functions. Additionally, far more data is needed to represent this ROM compared to the ROM based on the global basis.

This research presented a method that generalizes reduced order modelling of time and parameter dependent problems on sparse grids, by using multiple local bases instead of a fixed global basis to represent the underlying physics of a model. The novel local basis interpolation scheme was competitive with the global basis approach on test problems, showing that manifold methods
such as ITSGM have great potential to be utilized in reduced order models. However, more research on matrix interpolation methods is needed to improve the overall performance, scalability and efficiency of the algorithm.

The advantage of using protons to irradiate a tumour in cancer treatment, is that the energy can be delivered very precisely to the tumour without irradiating much of the surrounding tissue. The disadvantage of this is that small displacements of a patient can result in large deviations in the planned dose delivery according to the treatment plan. Therefore, many different dose distributions corresponding to different patient displacements have to be calculated to set up a robust treatment plan, which is computationally expensive. The solution to this problem is to use a combination of reduced order modelling and deep learning to perform computationally cheaper approximations of dose distributions corresponding to patient displacements. In previous research it has been shown that shallow and narrow neural networks were unable to predict the higher order modes of the reduced order model, which are required for the desired accuracy. Therefore, in this research it was studied how a feedforward neural network’s performance in predicting the higher order expansion coefficients of a reduced order model and the dose distributions depend on the neural network architecture and training approaches. \par First, a dose distribution was obtained by setting up a liver tumour patient treatment plan using treatment planning software matRad. Afterwards, dose distributions corresponding to simulated 1D, 2D and 3D displacements were generated using the optimized treatment plan. The amount of training samples was set to 1000, 2000 and 3000 and the amount of test samples was set to 200, 200 and 300 for the 1D, 2D and 3D case respectively. By using the singular value decomposition on the generated dose distributions of the training set, a reduced order model could be extracted and provided the expansion coefficients that represent the behaviour of the modes within the reduced order model. Afterwards, the expansion coefficients were inspected as a function of displacement to determine which mode showed strong non-linear behaviour. This mode was called the target mode or high order mode and all modes up until the target mode were included in the reduced basis that was used to approximate the dose distributions. Mode 18, 24 and 51 were chosen as the target modes for the 1D, 2D and 3D case respectively. Subsequently, the target accuracies of the dose predictions within the test set in terms of the average voxel fraction acceptances were calculated using a threshold of 1% difference between the true voxel dose and approximated dose, relative to the maximum dose of the true dose distribution. The target accuracies of the dose predictions within the test set were 99.86%, 99.64% and 99.46% for the 1D, 2D and 3D case respectively. In experiment 1, neural networks with varying amounts of hidden layers, neurons per hidden layer and loss functions were trained on the full set of training samples and, once trained, were used to predict the modes and dose distributions using the displacements from the test set. The amount of hidden layers and neurons per hidden layer were set to: 2, 3, 4 and 5 hidden layers with 40, 80, 120 and 160 neurons per hidden layer for the 1D-case; 2, 4 and 6 hidden layers with 100, 200, 300 and 400 neurons per hidden layer for the 2D-case; 2, 4 and 6 hidden layers with 150, 300, 450 and 600 neurons per hidden layer for the 3D-case. The chosen loss functions were the mean squared error and mean absolute error functions. Each hidden layer used the Rectified Linear Unit activation function and the Adamax optimizer was used. The highest achieved accuracies were 99.79%, 99.50% and 98.84% for the 1D, 2D and 3D case respectively. Experiment 2 was similar to experiment 1, but the networks for the 1D and 2D case were trained with half the amount of training samples. The neural networks of experiment 1 and 2 were trained with the displacements as input variables and all the corresponding expansion coefficients of the modes up until the target mode as output variables. In experiment 3, the neural networks of the 3D case were trained only with the target mode expansion coefficient instead of all expansion coefficients simultaneously. Finally, in experiment 4, a very wide but shallow neural network and a very deep but narrow neural network were compared with and without applying L2 regularization. The regularization parameter was set to 10^-3, 10^-4 and 10^-5. \par It was concluded that increasing the amount of hidden layers and neurons, up to a certain amount can improve the prediction of both the high order modes and dose distributions, after which the performance starts to deteriorate. The amount of hidden layers and neurons per layer at which this occurs is dependent on the loss function and the amount of training data. More precisely, the performance of neural networks that utilize the mean absolute error function deteriorate faster than networks utilizing the mean squared error loss function as the amount of hidden neurons and hidden layers increases. On the other hand, the amount of training data is also important to fully represent the features of the highly non-linear modes that deeper and wider neural networks are able to learn. Deeper networks trained longer than wider networks and regularization did not improve the performance of the neural networks within the trained amount of epochs and chosen amount of hidden layers and neurons per layer.