Local-Volatility Calibration with Hard-Constrained Differentiable Galerkin Layers

Local-volatility calibration is an ill-posed inverse parabolic problem, and standard pipelines typically separate surface interpolation from the pricing operator used downstream. We instead learn only the variance-rate coefficient $\kappa ={\sigma }_{\mathrm{loc}}^2/2$ and obtain prices from the exact log-moneyness Dupire equation through differentiable weighted-Galerkin layers. A positive parameterisation makes coefficient admissibility architectural; the weighted discretisation makes every implicit state matrix symmetric positive-definite (SPD); and reverse-mode gradients reuse the same sparse solves as the forward pass. Smoothness remains a statistical regulariser rather than a surrogate for either positivity or the PDE. We prove that finite soft penalties cannot certify membership in the admissible ellipticity set and test the distinction through synthetic recovery, a mechanism-level hard-versus-soft comparison, and out-of-sample calibration on an S&P 500 option snapshot. The results show that structural guarantees can be retained without claiming universal coefficient accuracy or a complete static-arbitrage certificate.