  Initial value problems -- a system of ordinary differential equations and
corresponding initial conditions -- can be used to describe many physical
phenomena including those arise in classical mechanics. We have developed a
novel approach to solve physics-based initial value problems using unsupervised
machine learning. We propose a deep learning framework that models the dynamics
of a variety of mechanical systems through neural networks. Our framework is
flexible, allowing us to solve non-linear, coupled, and chaotic dynamical
systems. We demonstrate the effectiveness of our approach on systems including
a free particle, a particle in a gravitational field, a classical pendulum, and
the H\'enon--Heiles system (a pair of coupled harmonic oscillators with a
non-linear perturbation, used in celestial mechanics). Our results show that
deep neural networks can successfully approximate solutions to these problems,
producing trajectories which conserve physical properties such as energy and
those with stationary action. We note that probabilistic activation functions,
as defined in this paper, are required to learn any solutions of initial value
problems in their strictest sense, and we introduce coupled neural networks to
learn solutions of coupled systems.
