Mesh Based Simulations with Spatial and Temporal awareness
For researchers in geometric deep learning and computational physics, this work addresses a critical bottleneck in training physics surrogates by incorporating numerical analysis principles, leading to more accurate and stable simulations.
The paper identifies that current training paradigms for GNN and Transformer surrogates in CFD rely on naive assumptions like node-wise supervision and explicit Euler time-stepping, which ignore stiff dynamics and flux continuity. They propose a unified framework with three innovations—Multi Node Prediction, Temporal Correction, and Geometric Inductive Biases—that consistently improves accuracy and stability across multiple architectures on diverse physics datasets, especially in long-horizon rollouts.
Machine Learning surrogates for Computational Fluid Dynamics (CFD), particularly Graph Neural Networks (GNNs) and Transformers, have become a new important approach for accelerating physics simulations. However, we identify a critical bottleneck in the field: while architectures have advanced significantly, the common underlying training paradigms remain bound to naive assumptions, such as node-wise supervision and explicit Euler time-stepping. These legacy choices ignore the stiff dynamics and local flux continuity inherent to numerous partial differential equations resolution methods, such as Finite Element, Difference, or Volume (FEM). In this work, we propose a unified framework to bridge the gap between geometric deep learning and rigorous numerical analysis. We introduce three key innovations: (1) Multi Node Prediction, a stencil-level objective that predicts field values for a node's full local topology, enforcing spatial derivative consistency; (2) Temporal Correction, replacing unstable explicit schemes with a predictor-corrector via temporal Cross-Attention; and (3) Geometric Inductive Biases, leveraging 3D Rotary Positional Embeddings (RoPE) to robustly capture rotational symmetries in unstructured meshes. We evaluate this framework across three architectures (MeshGraphNet, Transolver, and a Transformer) on diverse physics datasets. Our approach yields consistent improvements in accuracy and stability, particularly in long-horizon rollouts, while producing latent representations that generalize to unseen subtasks such as Wall Shear Stress or Pressure prediction. Code is available at https://github.com/DonsetPG/graph-physics.