Enforcing Dirichlet Boundary Conditions in Operator Learning

What the math gives to ML

The paper provides a hard architectural mechanism for satisfying homogeneous Dirichlet conditions without relying on boundary-penalty losses or boundary-heavy training data. Its transferable asset is the use of a geometry-dependent Laplacian eigenbasis: every hidden function is projected into the span of eigenfunctions that vanish on the boundary, so the constraint is preserved exactly on arbitrary Lipschitz domains and meshes. The most direct neural-network adaptation is to insert a differentiable truncated spectral projection after each operator layer, using finite-element eigenvectors for irregular domains and a learned or standard kernel-integral layer before projection.