Simulating and modeling the long-term dynamics of multi-object physical systems is an essential and challenging task. Current studies model the physical systems utilizing Graph Neural Networks (GNNs) with equivariant properties. Specifically, they model the dynamics as a sequence of discrete states with a fixed time interval and learn a direct mapping for all the two adjacent states. However, this direct mapping overlooks the continuous nature between the two states. Namely, we have verified that there are countless possible trajectories between two discrete dynamic states in current GNN-based direct mapping models. This issue greatly hinders the model generalization ability, leading to poor performance of the long-term simulation. In this paper, to better model the latent trajectory through discrete supervision signals, we propose a Physics-Inspired Neural Graph ODE (PINGO) algorithm. In PINGO, to ensure the uniqueness of the trajectory, we construct a Physics-Inspired Neural ODE framework to update the latent trajectory. Meanwhile, to effectively capture intricate interactions among objects, we use a GNN-based model to parameterize Neural ODE in a plug-and-play manner. Furthermore, we prove that the discrepancy between the learned trajectory of PIGNO and the true trajectory can be theoretically bounded. Extensive experiments verify our theoretical findings and demonstrate that our model yields an order-of-magnitude improvement over the state-of-the-art baselines, especially on long-term predictions and roll-out errors.

翻译：模拟和建模多物体物理系统的长期动力学是一项重要且具有挑战性的任务。当前研究利用具有等变属性的图神经网络（GNN）对物理系统进行建模。具体而言，他们将动力学建模为固定时间间隔的离散状态序列，并学习所有相邻两个状态之间的直接映射。然而，这种直接映射忽略了两个状态之间的连续性质。即，我们已验证在现有基于GNN的直接映射模型中，两个离散动力学状态之间存在无数可能的轨迹。这一问题极大地阻碍了模型的泛化能力，导致长期模拟性能不佳。本文为通过离散监督信号更好地对潜在轨迹进行建模，提出了一种物理启发的神经图常微分方程（PINGO）算法。在PINGO中，为确保轨迹的唯一性，我们构建了一个物理启发的神经ODE框架来更新潜在轨迹。同时，为有效捕捉物体间的复杂交互，我们使用基于GNN的模型以即插即用的方式对神经ODE进行参数化。此外，我们证明了PIGNO学习轨迹与真实轨迹之间的差异在理论上是有界的。大量实验验证了我们的理论发现，并表明我们的模型在长期预测和滚动误差方面，相较于最先进的基线方法实现了数量级的改进。