Precision Physics in Extensions of the Standard Model

The Standard Model of particle physics has been remarkably successful at explaining the behaviour of nature at very high energies. It has been thoroughly tested by experiments at the Large Hadron Collider and almost all of its predictions have agreed closely with observations. Despite this, there are many phenomena that it cannot explain, such as the origins of neutrino masses. Therefore, the Standard Model alone cannot provide a complete explanation of reality, and so there must exist physics beyond it. However, finding it is particularly difficult due to the aforementioned success of the Standard Model. In this thesis, we continue the search for this new physics by carrying out five separate studies that could contribute towards the overall goal. We calculate new limits on specific extensions of the Standard Model by using lepton-flavour-violating decays of $\tau$ leptons. We also use electroweak precision measurements for the first time to constrain the trilinear self-coupling of the Higgs boson, and perform the most precise calculation to date of the production of a heavy neutrino via gluon fusion. Considering machine-learning techniques, we improve the robustness of an autoencoder used for unsupervised searches for new physics, and we develop a new approach to using neural networks for solving differential equations, which we apply to the calculation of cosmological phase transitions. A general theme across all of these investigations is the importance of a high precision, both in terms of the theoretical calculations, and the experiments through which they are tested.