Mesh Based Simulations with Spatial and Temporal awareness

arXiv cs.LG / 5/5/2026

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Key Points

The paper argues that ML surrogate models for CFD (notably GNNs and Transformers) are still hampered by training paradigms that use naive node-wise supervision and explicit Euler time-stepping, which fail to respect stiff dynamics and local flux continuity in many PDE discretizations.
It proposes a unified, physics-aware framework combining three innovations: a stencil-level multi-node prediction objective for spatial derivative consistency, a temporal correction method using a predictor-corrector with temporal cross-attention, and geometric inductive biases via 3D RoPE for unstructured meshes.
The framework is evaluated on multiple architectures (MeshGraphNet, Transolver, and a Transformer) across several physics datasets, showing improved accuracy and stability, especially for long-horizon rollouts.
The learned latent representations also generalize to related unseen tasks such as Wall Shear Stress and pressure prediction, indicating broader usefulness beyond the training targets.
The authors release accompanying code on GitHub, enabling others to reproduce and build on the approach.

Abstract

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.