In this thesis, we propose HyperVerlet, a novel self-supervised deep learning method for numerically solving initial value problems of Hamiltonian systems. We provide theoretical proof of a lower local and global truncation error compared to the prevalent velocity Verlet. To gather empirical evidence, HyperVerlet is tested on dynamical systems where conservative quantities such as energy and total momentum is of high importance including an undamped spring-mass, an idealized pendulum, and the chaotic three-body spring-mass systems. Empirical evidence shows that HyperVerlet outperforms velocity Verlet, other hypersolvers, and in some cases perform on par with 4th-order solvers. Depending on the choice of the neural network-based corrector, HyperVerlet has the capability of being both a symplectic and non-symplectic solver, which is a geometric property characterizing volume-preservation, making it ideal of long duration simulations.