  Realistic models of physical world rely on differentiable symmetries that, in
turn, correspond to conservation laws. Recent works on Lagrangian and
Hamiltonian neural networks show that the underlying symmetries of a system can
be easily learned by a neural network when provided with an appropriate
inductive bias. However, these models still suffer from issues such as
inability to generalize to arbitrary system sizes, poor interpretability, and
most importantly, inability to learn translational and rotational symmetries,
which lead to the conservation laws of linear and angular momentum,
respectively. Here, we present a momentum conserving Lagrangian neural network
(MCLNN) that learns the Lagrangian of a system, while also preserving the
translational and rotational symmetries. We test our approach on linear and
non-linear spring systems, and a gravitational system, demonstrating the energy
and momentum conservation. We also show that the model developed can generalize
to systems of any arbitrary size. Finally, we discuss the interpretability of
the MCLNN, which directly provides physical insights into the interactions of
multi-particle systems.
