GLOBE: Accurate and Generalizable PDE Surrogates using Domain-Inspired Architectures and Equivariances

Abstract

We introduce GLOBE, a new neural surrogate for homogeneous PDEs that draws inductive bias from boundary-element methods and equivariant ML. GLOBE represents solutions as superpositions of learnable Green's-function-like kernels evaluated from boundary faces to targets, composed across multiscale branches and communication hyperlayers. The architecture is translation-, rotation-, and parity-equivariant; discretization-invariant in the fine-mesh limit; and units-invariant via rigorous nondimensionalization. An explicit far-field decay envelope stabilizes extrapolation, boundary-to-boundary hyperlayer communication mediates long-range coupling, and the all-to-all boundary-to-target evaluation yields a global receptive field that respects PDE information flow, even for elliptic PDEs. On AirFRANS (steady incompressible RANS over NACA airfoils), GLOBE achieves substantial accuracy improvements. On the "Full" split, it reduces mean-squared error by roughly 200x on all fields relative to the dataset's reference baselines, and roughly 50x relative to the next-best-performing model. In the "Scarce" split, it achieves over 100x lower error on velocity and pressure fields and over 600x lower error on surface pressure than Transolver. Qualitative results show sharp near-wall gradients, coherent wakes, and limited errors under modest extrapolation in Reynolds number and angle of attack. In addition to this accuracy, the model is quite compact (117k parameters), and fields can be evaluated at arbitrary points during inference. We also demonstrate the ability to train and predict with non-watertight meshes, which has strong practical implications. These results show that rigorous physics- and domain-inspired inductive biases can achieve large gains in accuracy, generalizability, and practicality for ML-based PDE surrogates for industrial computer-aided engineering (CAE).