The present project focuses on modelling and development of modern computational tools required for investigating forefront open problems within the interplay of computational mathematics with atomic, nuclear and non-standard particle physics. Its realization is based on the collaboration of computer science, muon physics as well as particle physics experts and the employment of modern optimization approaches to build the required algorithms. Specifically, the main aim is the solution of the differential Dirac and Breit-Darwin equations by adopting neural network techniques and other schemes in developing efficient codes for high sensitivity predictions in order to facilitate interpretations of appreciably high precision experimental data related to muon physics and exotic purely leptonic atoms.