  Neural networks with physics based inductive biases such as Lagrangian neural
networks (LNN), and Hamiltonian neural networks (HNN) learn the dynamics of
physical systems by encoding strong inductive biases. Alternatively, Neural
ODEs with appropriate inductive biases have also been shown to give similar
performances. However, these models, when applied to particle based systems,
are transductive in nature and hence, do not generalize to large system sizes.
In this paper, we present a graph based neural ODE, GNODE, to learn the time
evolution of dynamical systems. Further, we carefully analyse the role of
different inductive biases on the performance of GNODE. We show that, similar
to LNN and HNN, encoding the constraints explicitly can significantly improve
the training efficiency and performance of GNODE significantly. Our experiments
also assess the value of additional inductive biases, such as Newtons third
law, on the final performance of the model. We demonstrate that inducing these
biases can enhance the performance of model by orders of magnitude in terms of
both energy violation and rollout error. Interestingly, we observe that the
GNODE trained with the most effective inductive biases, namely MCGNODE,
outperforms the graph versions of LNN and HNN, namely, Lagrangian graph
networks (LGN) and Hamiltonian graph networks (HGN) in terms of energy
violation error by approx 4 orders of magnitude for a pendulum system, and
approx 2 orders of magnitude for spring systems. These results suggest that
competitive performances with energy conserving neural networks can be obtained
for NODE based systems by inducing appropriate inductive biases.
