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The solution of time dependent differential equations with neural networks
has attracted a lot of attention recently. The central idea is to learn the
laws that govern the evolution of the solution from data, which might be
polluted with random noise. However, in contrast to other machine learning
applications, usually a lot is known about the system at hand. For example, for
many dynamical systems physical quantities such as energy or (angular) momentum
are exactly conserved. Hence, the neural network has to learn these
conservation laws from data and they will only be satisfied approximately due
to finite training time and random noise. In this paper we present an
alternative approach which uses Noether's Theorem to inherently incorporate
conservation laws into the architecture of the neural network. We demonstrate
that this leads to better predictions for three model systems: the motion of a
non-relativistic particle in a three-dimensional Newtonian gravitational
potential, the motion of a massive relativistic particle in the Schwarzschild
metric and a system of two interacting particles in four dimensions.
