Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics

We introduce a meshfree exterior calculus (MEEC) for learning structure-preserving descriptions of physics on point clouds, and use it to build MEEC-Net, a data-efficient surrogate that transfers across resolutions, geometries, and physical parameters. MEEC equips an $\varepsilon$-ball graph with virtual node and edge measures via a single sparse Schur complement solve; the resulting complex satisfies discrete conservation exactly, is end-to-end differentiable in the point positions, and exposes a direct geometry-to-physics link without the mesh-generation step required by conventional structure-preserving discretizations. MEEC-Net learns unknown physics as a shared edge-wise flux law in an SO($d$)-invariant local frame, so the same kernel produces compatible fluxes on any point cloud whose features lie in the training range. We prove a solution-error bound that splits into discretization and kernel-approximation terms which is independent of problem geometry, explaining the observed transfer from very few examples. We show that single-solution training transfers to unseen geometries, boundary conditions, and physical parameters. On five canonical PDE benchmarks MEEC-Net achieves 1-2 orders of magnitude lower out-of-distribution error than baseline neural-operator approaches. On the SimJEB structural-bracket benchmark it achieves competitive error while using substantially fewer training geometries.

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.

Neural surrogate models for physical simulations are trained on discretized samples of continuous domains, where the induced empirical measure leads to uneven supervision, biasing optimization and causing spatial inconsistencies in physical fidelity. To mitigate this measure-induced bias, we propose M$^3$ (Multi-scale Morton Measure), a scalable framework that balances training measures by partitioning space according to physical variation and allocating supervision across multiple scales. Applied to three industrial-scale datasets with diverse discretizations, M$^3$ consistently improves predictions in the continuous physical domain, achieving up to 4.7$\times$ lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M $\rightarrow$ 16M $\rightarrow$ 1.6M points), where M$^3$-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative $L_2$ error by 3--4$\times$ and the corresponding MSE by up to 13$\times$. These results highlight data distribution as a key factor in operator learning and position M$^3$ as a scalable, data-efficient approach for physically consistent modeling. Code is available at https://github.com/PhysDataRefine/M3.