Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator

What the math gives to ML

The paper provides a constructive recipe for representing operators on changing surfaces in one fixed function space: a surface contravariant Piola map transports geometry-dependent Maxwell trace spaces to a reference space, while Jacobian factors cancel in the pulled-back variational form. Its second transferable asset is dimension-independent sparse approximation: if shape perturbation amplitudes belong to \(\ell^p\), \(0<p<1\), holomorphic parameter dependence implies rapidly decaying multivariate Legendre coefficients. Together these suggest geometry-conditioned neural operators or hypernetworks that reuse one mesh and one parameter indexing across shapes, while replacing dense dependence on infinitely many shape variables by a learned sparse polynomial dependence. The most credible first test is a fixed-reference neural operator for PDE or scattering data across randomly deformed domains, compared against remeshing and dense parameter encodings.