This code contains an experimental architecture called GLOBE: Green's-function-Like Operator for Boundary Element PDEs.

Paper: https://arxiv.org/abs/2511.15856

GLOBE is intended to be used for PDEs with the following mathematical properties:

• The important behavior of the PDE is primarily driven by the boundary conditions - in particular, the geometry of the boundary.
• The PDE is elliptic, requiring global information propagation to solve correctly.
• The engineering quantities of interest are primarily located on or near the boundary.
• The PDE is either linear or can be roughly-approximated as linear for significant portions of the domain.

A GLOBE model represents the solution of a PDE as a linear combination of learnable kernel functions evaluated from boundary source faces to target points.

The architecture itself has many invariant properties. So, without retraining the model, you can do the following:

• Translation-equivariant: If the problem setup is translated the predictions exactly follow the translation.
• Rotation-equivariant: If the problem setup is rotated the predictions exactly follow the rotation.
  • Also invariant to in-plane rotations of any given face on the boundary mesh.
• Discretization-invariant: If you decimate the boundary mesh uniformly, the predictions do not change in the fine-mesh limit.
• Parity-equivariant: If you reflect the problem across a plane, the predictions exactly follow the reflection. Also, a problem with a plane of symmetry in the inputs will yield predictions that also have a plane of symmetry.
• Units-invariant: If you pose the same physical problem in a different scale (e.g., using different units), the predictions will be the same if you use the reference_lengths argument appropriately.

In addition:

• The model bakes in a physical prior that the influence of any individual small feature tends to fall off as 1/r^2 in 3D and 1/r in 2D. Model is globally interacting but with provably-diminishing influence as a function of distance -- this describes many elliptic PDEs of industrial relevance.

• Notably, the model is not invariant with respect to normal-vector-flips of the boundary mesh - a face with its normal facing inwards and a face with its normal facing outwards are encoded differently, and hence, will yield (in general) different predictions. You should ensure that the boundary mesh is consistently oriented between cases, and also that individual faces are consistently oriented within a single case. (This is by design, not by accident - if the model is used to encode oriented boundary conditions, like Neumann conditions, normal direction matters.)
• Requires Python 3.12+ if you want to use the torch.compile decorator. This is because GLOBE extensively uses Python built-in dataclasses with @cached_property decorators - in previous versions of Python, these acquired a threading.RLock to guard first-time computation. This RLock is not traceable by Torch-Dynamo, and so, torch.compile will fail. This was fixed in Python 3.12, which made further thread-safety enhancements to cached_property that eliminated the need for the RLock.