Grass-covered dikes are a widely used measure for flood prevention. Overtopping waves can cause erosion that poses significant risks for flood safety, especially with rising sea levels and increasing storm intensities. During storm events, which involves frequent wave overtopping, erosion occurs progressively as multiple individual waves flow over the dike. This causes the dike profile to change over time as erosion holes begin to appear, which can influence the erosion potential of subsequent overtopping waves.

Current methods to estimate forces and erosion from overtopping waves rely on numerical models or analytical and empirical formulas. Numerical models are generally applicable and provide detailed spatial and temporal results, but are computationally too expensive to simulate an entire storm whilst updating the bed profile. Formulas offer quick results but lack general applicability and fail to account for the effects of erosion holes. No current method efficiently combines computational speed, detailed results and broad applicability, making it challenging to study the effects of gradually increasing erosion on overtopping waves.

This research focuses on the development of a new method that provides rapid and reliable wave overtopping simulations that can integrate erosion. This is done through surrogate modelling, which aims to create a computationally cheap model that emulates a detailed model. The foundation of this surrogate model is a Transformer-based deep learning architecture, which has proven superior in handling spatiotemporal processes.

The surrogate model is created by adapting the Vision Transformer model into a new model that can perform next-frame prediction, which involves the prediction of a next frame based on a sequence of input frames. The developed model can take an input sequence, recognize spatial and temporal patterns, and project them into a predicted future timestep. The model is trained on a dataset of overtopping wave simulations produced by the CFD software OpenFOAM. A masked loss function is applied to enhance the training process by forcing the model to focus on improving the relevant errors.

Using the trained model to generate wave overtopping simulations showed that it can accurately replicate the original CFD simulations. The surrogate model is validated against the original CFD simulations by comparing maximum values and time series for flow velocity and water depth at four different locations along the dike. The results generally show good agreement in capturing the maximum values, as well as the time of arrival and the overtopping duration. Simulating wave overtopping over an eroded dike profile showed promising results, though performance could improve with a larger and more diverse training dataset.

This research demonstrates that a Transformer-based surrogate model can effectively emulate wave overtopping simulations produced by CFD software. The surrogate model's speed and simplicity enables the simulation of a storm and updating of the bed profile. The model has not reached its full potential due to limitations in the training dataset and nuances in the simulation technique that require further refinement. However, it serves as a proof of concept that this surrogate model can provide a new tool in wave overtopping modelling, creating possibilities for new studies such as the influence of erosion on overtopping waves during storm conditions, or for probabilistic calculations that require a large number of simulations.