Learning PDE-Feasible Design Manifolds for Solver-Amortized Optimization

PDE-constrained optimization repeatedly solves a design-dependent state equation. For a class of separable elliptic PDEs, we show that one reference solution induces coefficient–state pairs that preserve compatible source and boundary data and satisfy the residual-inheritance identity $R_{\theta }=c_{\theta }{R}_0$. An exact reference yields a family of exact states, while an approximate reference transfers an explicit residual certificate. This enables hard-constrained design updates from cached fields without a new state solve. Because a fixed family spans only restricted directions, a full-space check at restricted stationarity constructs a bound- and volume-constrained Wendland-RBF candidate from the inaccessible gradient component; one candidate solve and an acceptance gate decide whether to update the reference. Across thermal experiments, analytical updates take $0.12-0.14$ s, versus $165-493$ s for independent PINN re-solves. In end-to-end physics-informed optimization, our method trains $3$ state PINNs rather than $18$ and reaches a $51.7\%$ lower FVM-audited resistance. The framework therefore amortizes expensive state solves through residual-certified reuse.