  Invariants and conservation laws convey critical information about the
underlying dynamics of a system, yet it is generally infeasible to find them
without any prior knowledge. We propose ConservNet to achieve this goal, a
neural network that extracts a conserved quantity from grouped data where the
members of each group share invariants. As a neural network trained with a
novel and intuitive loss function called noise-variance loss, ConservNet learns
the hidden invariants in each group of multi-dimensional observables in a
data-driven, end-to-end manner. We demonstrate the capability of our model with
simulated systems having invariants as well as a real-world double pendulum
trajectory. ConservNet successfully discovers underlying invariants from the
systems from a small number of data points, namely less than several thousand.
Since the model is robust to noise and data conditions compared to baseline,
our approach is directly applicable to experimental data for discovering hidden
conservation laws and relationships between variables.
