Mesh Based Simulations with Spatial and Temporal awareness

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 \url{https://github.com/DonsetPG/graph-physics}.

Engineers use computer simulations to predict how fluids flow or materials deform, but these simulations can be slow because they solve physics equations over many small connected pieces called a mesh. Machine learning can speed this up by learning from previous simulations, but common AI simulators often predict one mesh point at a time and advance time with a simple update rule. This can miss how neighboring points affect each other and can make errors grow during long simulations. We propose three changes that make these AI simulators more aware of local shape and time. First, during training, the model must predict not only what happens at one point, but also what happens nearby, encouraging local physical consistency. Second, the model adds a correction step that checks a predicted future state against the current state before moving forward. Third, it uses 3D position information so models better understand direction and orientation on irregular meshes. Tested on fluid flow around a cylinder, blood flow in a brain aneurysm, and a deforming plate, the approach reduced long-term prediction errors by about 20–30% with modest extra computation. This could make learned physics simulators more accurate and reliable for engineering and biomedical applications.