Discovering Message Passing Hierarchies for Mesh-Based Physics Simulation

Graph neural networks have emerged as a powerful tool for large-scale mesh-based physics simulation. Existing approaches primarily employ hierarchical, multi-scale message passing to capture long-range dependencies within the graph. However, these graph hierarchies are typically fixed and manually designed, which do not adapt to the evolving dynamics present in complex physical systems. In this paper, we introduce a novel neural network named DHMP, which learns Dynamic Hierarchies for Message Passing networks through a differentiable node selection method. The key component is the anisotropic message passing mechanism, which operates at both intra-level and inter-level interactions. Unlike existing methods, it first supports directionally non-uniform aggregation of dynamic features between adjacent nodes within each graph hierarchy. Second, it determines node selection probabilities for the next hierarchy according to different physical contexts, thereby creating more flexible message shortcuts for learning remote node relations. Our experiments demonstrate the effectiveness of DHMP, achieving 22.7% improvement on average compared to recent fixed-hierarchy message passing networks across five classic physics simulation datasets. Simulating physical systems with deep neural networks has achieved remarkable success due to their efficiency compared with traditional numerical solvers. Graph Neural Networks (GNNs) have been validated as a powerful tool for mesh-based physical scenarios, such as fluids and rigid collisions (Wu et al., 2020). The primary mechanism driving the GNN-based models is message passing (Sanchez-Gonzalez et al., 2020; Pfaff et al., 2021; Allen et al., 2023).