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

Neural networks with physics-based inductive biases such as Lagrangian neural networks (LNNs), and Hamiltonian neural networks (HNNs) 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 analyze 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 Newton’s third law, on the final performance of the model. We demonstrate that inducing these biases can enhance the performance of the 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 ∼4 orders of magnitude for a pendulum system, and ∼2 orders of magnitude for spring systems. These results suggest that NODE-based systems can give competitive performances with energy-conserving neural networks by employing appropriate inductive biases.