GLOBE: Accurate Surrogates for Boundary-Driven PDEs via Domain-Inspired Architectures and Equivariance

We introduce GLOBE, a neural surrogate for boundary-driven, homogeneous, weakly nonlinear PDEs that draws inductive bias from boundary-element methods and equivariant ML, representing solutions as superpositions of learnable Green's-function-like kernels evaluated from boundary faces to targets. The architecture is translation-, rotation-, and parity-equivariant; discretization-invariant; and units-invariant via rigorous nondimensionalization. On AirFRANS (steady incompressible RANS over NACA airfoils), GLOBE reduces mean-squared error by roughly $200\times$ relative to dataset baselines and $50\times$ relative to the next-best ML surrogate on interpolation tasks, with $7\times$–$70\times$ improvements over the next-best surrogate in low-data regimes. The compact ${\sim} 117 \rm k$-parameter model evaluates fields at arbitrary points, handles non-watertight meshes, and generalizes under Reynolds number and angle-of-attack distribution shifts. We further introduce a hierarchical Barnes-Hut-style acceleration that exploits the kernel's guaranteed decay of long-range influences to reduce evaluation cost from quadratic to near-linear in the number of boundary elements, with a tunable accuracy parameter that recovers exact dense evaluation in its strict limit. We demonstrate scalability to 3D industrial geometries via a proof-of-concept on DrivAerML car aerodynamics. Together these results show that physics-inspired inductive biases yield large gains in accuracy and practicality for ML-based surrogates of boundary-driven PDEs while remaining tractable at industrial 3D scales.