Geometry-aware LegONet for PDE Learning on Arbitrary Domains

What the math gives to ML

The paper's transferable asset is an algebraic interface between reusable ambient operator blocks and a new constrained geometry: represent every admissible state as a particular boundary-satisfying state plus a mass-orthonormal nullspace coordinate. This avoids the generally incorrect practice of evolving an unconstrained state and projecting it back after each neural or numerical step, because the reduced vector field is evaluated directly on the tangent space. A practical neural analogue is a hard-constraint operator module that freezes mechanism networks in ambient coordinates while learning or recomputing only a geometry-specific nullspace basis and coordinate maps.