Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.

翻译：近年来，图神经网络因其归纳特性带来的零样本泛化能力，在动力系统模拟领域受到广泛关注。同样地，深度学习框架中融入物理信息的归纳偏置已被证明在物理系统动力学学习方面具有卓越性能。越来越多研究尝试将这两种方法相结合。本文系统评估了十三种不同图神经网络（包括哈密顿图神经网络、拉格朗日图神经网络、图神经ODE及其显式约束与不同架构变体）的性能。我们简要阐释了这些系统在归纳偏置与图架构方面的理论表述，着重说明其相似性与差异性。通过弹簧、摆锤、引力场及三维可变形固体等动力系统模型，我们从滚动误差、能量与动量等守恒量及对未训练系统规模的泛化能力三个维度进行性能对比。研究表明，具有额外归纳偏置（如显式约束、动能与势能解耦）的图神经网络展现出显著增强的性能。此外，所有物理信息图神经网络均能在训练系统规模扩大一个数量级时保持零样本泛化能力，这为模拟大规模真实系统提供了可行路径。