EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions

TL;DR

Fixed graph hierarchies limit a model's ability to adapt to changing physical dynamics. EvoMesh (DHMP) introduces time-evolving, context-aware graph hierarchies and anisotropic message passing to capture directional dependencies across scales, using differentiable node selection via Gumbel-Softmax and learnable inter-level propagation. Key components include edge-importance weights $\alpha_{ij}$, DiffSELECT for adaptive downsampling with $\mathbf{z}_i^l = \text{Gumbel-Softmax}(\log \pi_{i,0}^l, \log \pi_{i,1}^l)$, and REDUCE/EXPAND with a FeatureMixing fusion, enabling robust long-range dynamics modeling. Empirical results across five benchmarks show EvoMesh outperforms fixed-hierarchy methods and generalizes to evolving meshes and varying resolutions, offering scalable, accurate physics simulations on complex, time-varying systems.

Abstract

Graph neural networks have been a powerful tool for mesh-based physical simulation. To efficiently model large-scale systems, existing methods mainly employ hierarchical graph structures to capture multi-scale node relations. However, these graph hierarchies are typically manually designed and fixed, limiting their ability to adapt to the evolving dynamics of complex physical systems. We propose EvoMesh, a fully differentiable framework that jointly learns graph hierarchies and physical dynamics, adaptively guided by physical inputs. EvoMesh introduces anisotropic message passing, which enables direction-specific aggregation of dynamic features between nodes within each hierarchy, while simultaneously learning node selection probabilities for the next hierarchical level based on physical context. This design creates more flexible message shortcuts and enhances the model's capacity to capture long-range dependencies. Extensive experiments on five benchmark physical simulation datasets show that EvoMesh outperforms recent fixed-hierarchy message passing networks by large margins. The project page is available at https://hbell99.github.io/evo-mesh/.

Figures (10)

• Figure 1: The architecture of DHMP. Physical dynamics is modeled on multiple graph resolutions with adaptive structures, $\mathcal{G}_1, \mathcal{G}_2, \ldots, \mathcal{G}_L$, and are processed using their respective AMP layers. The DiffSELECT operation performs differentiable pooling to create coarser graphs with learnable downsampling probabilities. REDUCE and EXPAND integrate inter-level information using learned feature aggregation weights over the neighboring nodes. DHMP is trained end-to-end with one-step supervision.
• Figure 2: Prediction showcases over $\textbf{400}$ future steps on CylinderFlow. From the displayed error maps, it is evident that DHMP effectively captures long-term dynamics, providing predictions that closely align with the ground truth.
• Figure 3: A demonstration of how the learned hierarchies adapt to evolving physical dynamics.Top: the velocity field from the true data. Bottom: the temporal difference of the velocity fields between adjacent time steps alongside the constructed coarser-level mesh graph ($\mathcal{G}_{l=4}$). The highlighted areas demonstrate a notable experimental phenomenon: the mesh dynamically evolves with the data context and aligns with the critical areas of change in the data.
• Figure 4: Ablation studies. We provide analyses of time-evolving hierarchies, anisotropic intra-level propagation, and learnable inter-level feature propagation. The red dashed lines represent results from BSMS-GNN cao2023efficient. Lower values indicate better performance.
• Figure 5: A demonstration of how the predicted anisotropic edge weights respond to dramatic changes in physical quantities over time.Top: Visualizations of the variance in the generated anisotropic weights, calculated on adjacent edges. Bottom: Variance in physical quantities over time. The strong correlation between them highlights the AMP's ability to detect significant patterns in data.