A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

What the math gives to ML

The paper provides a constructive shape-sensitivity law for a nonlinear potential whose Monge–Ampère measure has an interface-supported component. Its transferable asset is not the Kähler setting itself, but the combination of a boundary flux trace, a positive geometric boundary measure, and a Hadamard derivative proportional to the squared normal flux. This suggests a principled interface loss or shape-gradient signal for neural implicit free-boundary solvers, especially when the learned potential is nonsmooth and ordinary pointwise boundary derivatives are unreliable. The first implementation should use an implicit surface and neural potential, estimate the flux and mixed Monge–Ampère weights at boundary quadrature points, and test whether the resulting shape update agrees with finite-difference energy changes.