Abstract:

Graph Neural Network （GNN） is increasingly applied to complex fluid system predictions due to the superior capability in handling structured grids and strong combinatorial generalization. However， from a Lagrangian mesh-free perspective， GNN has unpredictable output variations when processing fluid particle information subjected to translation， rotation， or reflection transformations. To address this problem， an Equivariant Graph Neural Network-based Simulator （EGNS） method was proposed. Geometric vectors were first converted into relative equivariants. Then， equivariant message passing was employed at each step to ensure equivariance of entire neural network， maintaining consistency of spatial transformations between output and input equivariants. Finally， the optimized EGNS model was obtained by training in particle trajectories simulated with the Smoothed Particle Hydrodynamics （SPH） method. Experimental results on public fluid simulation datasets demonstrate that EGNS has superior predictive performance compared with Graph Neural Network-based Simulator （GNS）； specifically， EGNS achieves better accuracy in predicting fluid particle movement， velocity， and typical details， decreasing Mean Squared Error （MSE） in predicting particle positions by about 16%.

Key words: equivariant graph neural network, equivariance, smoothed particle hydrodynamics, fluid particle, flow prediction

摘要：

图神经网络（GNN）因能较好解决结构网格的问题，且有较强的组合泛化能力，被越来越多地应用于复杂的流体系统预测。然而，在拉格朗日无网格视角下，经过平移、旋转或翻转变换的流体粒子信息输入GNN会出现不可预测的输出问题。为了解决该问题，提出基于等变图神经网络模拟（EGNS）的方法。首先，将几何向量转换为相对的等变量；其次，通过每一步具有等变性的消息传递使整个神经网络具有等变性，网络输出与输入等变量的空间变换保持一致；最后，在光滑粒子流体动力学（SPH）方法模拟的粒子轨迹里训练得到较优的EGNS模型。在公开流体仿真数据集上的实验结果表明，EGNS具有良好预测效果，相较于图神经网络模拟（GNS）的方法，EGNS在流体粒子运动形态、速度及典型细节的表现力上更准确，预测粒子的位置均方误差（MSE）减小了约16%。

关键词: 等变图神经网络, 等变性, 光滑粒子动力学, 流体粒子, 流动预测