Learning dynamical systems that respect physical symmetries and constraints remains a fundamental challenge in data-driven modeling. Integrating physical laws with graph neural networks facilitates principled modeling of complex N-body dynamics and yields accurate and permutation-invariant models. However, training graph neural networks with iterative, gradient-descent-based optimization algorithms (e.g., Adam, RMSProp, LBFGS) often leads to slow training, especially for large, complex systems. In comparison to 15 different optimizers, we demonstrate that Hamiltonian Graph Networks (HGN) can be trained 150-600x faster - but with comparable accuracy - by replacing iterative optimization with random feature-based parameter construction. We show robust performance in diverse simulations, including N-body mass-spring and molecular dynamics systems in up to dimensions and 10,000 particles with different geometries, while retaining essential physical invariances with respect to permutation, rotation, and translation. Our proposed approach is benchmarked using a NeurIPS 2022 Datasets and Benchmarks Track publication to further demonstrate its versatility. We reveal that even when trained on minimal 8-node systems, the model can generalize in a zero-shot manner to systems as large as 4096 nodes without retraining. Our work challenges the dominance of iterative gradient-descent-based optimization algorithms for training neural network models for physical systems.

翻译：学习遵循物理对称性和约束条件的动态系统仍然是数据驱动建模中的一个基本挑战。将物理定律与图神经网络相结合，有助于对复杂N体动力学进行原理性建模，并产生准确且置换不变的模型。然而，使用基于梯度下降的迭代优化算法（如Adam、RMSProp、LBFGS）训练图神经网络通常会导致训练速度缓慢，特别是在处理大型复杂系统时。与15种不同优化器相比，我们证明通过用基于随机特征的参数构建替代迭代优化，哈密顿图网络（HGN）的训练速度可提升150-600倍，同时保持相当的精度。我们在多种仿真中展示了稳健的性能，包括N体质量弹簧系统和分子动力学系统，这些系统可扩展至高达10,000个粒子并具有不同几何构型，同时保持对置换、旋转和平移的基本物理不变性。我们使用NeurIPS 2022数据集与基准测试轨道的出版物对所提方法进行基准评估，进一步证明了其通用性。研究发现，即使在仅使用8节点系统进行训练的情况下，该模型也能以零样本方式泛化至多达4096节点的系统而无需重新训练。我们的工作对基于迭代梯度下降的优化算法在物理系统神经网络模型训练中的主导地位提出了挑战。