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

TL;DR

GLOBE introduces a boundary-element-inspired, equivariant neural surrogate for homogeneous PDEs, achieving global information flow through all-to-all boundary interactions and boundary-to-interior evaluation. By enforcing translation-, rotation-, and parity-equivariance, discretization- and units-invariance, and a principled far-field decay, the model delivers strong generalization and robustness, including to non-watertight meshes. The architecture combines Padé-approximant MLPs, multiscale kernel branches, and communication hyperlayers to mimic Green's-function behavior while remaining data-efficient. Evaluated on AirFRANS, GLOBE delivers orders-of-magnitude improvements over state-of-the-art surrogates in interpolation, low-data, and extrapolation settings, with physically meaningful nondimensional metrics and robust qualitative predictions. The results advocate for physics-informed architectural design as a path to reliable, industrially applicable ML PDE surrogates, with clear avenues for 3D extension and broader PDE applicability.

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).

Figures (16)

• Figure 1: The input-output mapping of a GLOBE model.
• Figure 2: GLOBE architecture. The model operates at three hierarchical levels. Top: Communication hyperlayers propagate latent information between boundary partitions before final evaluation. Middle: Each hyperlayer contains multiscale kernels operating at different reference length scales, with outputs summed. Bottom: Individual kernels evaluate all-to-all source-to-target influences through invariant feature engineering, Pad√©-approximant networks, far-field decay envelopes, and equivariant vector reprojection, then aggregate over sources weighted by strengths and face areas.
• Figure 3: The $\operatorname{smoothlog}(x)$ function and its derivative, with near- and far-field asymptotic limits shown as dashed lines.
• Figure 4: GLOBE model predictions on the first validation sample of the Full split. Rows show ground truth, GLOBE predictions, and error, respectively. Each column corresponds to a different field of interest. For vector fields, the magnitude is shown.
• Figure 5: GLOBE model predictions on the first validation sample of the Scarce split. Rows show ground truth, GLOBE predictions, and error, respectively. Each column corresponds to a different field of interest. For vector fields, the magnitude is shown. Even with only 200 training samples, GLOBE results on validation cases qualitatively remain sharp and physically consistent.