Enhancing the Inductive Biases of Graph Neural ODE for Modeling Dynamical Systems

Abstract: 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.