Certifying Physics-Informed Surrogates for PDE-Constrained Optimization

Physics-informed neural surrogates can reduce the cost of PDE-constrained optimization, but an optimizer itself can drive iterates into regions where the surrogate is least reliable. We introduce SA-CPITR, a solver-assisted certified trust-region method that treats a learned PDE surrogate as an inexact optimization oracle whose errors are bounded online. A posteriori majorants convert PDE residual information into computable error bounds for the state and sensitivities, which we assemble into certificates for the reduced objective and its gradient. Since the majorants hold for any conforming candidate, the same formulas certify both the neural surrogate and the high-fidelity solver outputs. These certificates control trust-region acceptance and termination, guaranteeing objective descent for every accepted step without evaluating the true objective. When a certificate becomes insufficient, the method spends state or gradient high-fidelity solves to generate a reusable anchor that tightens subsequent certificates. If the target tolerance cannot be certified within the anchor event budget, the algorithm returns its best certified bound rather than an uncertified solution. We prove certificate validity, guaranteed descent, and convergence to first-order criticality under explicit assumptions. On five PDE control problems, from linear elliptic to nonlinear and hyperbolic classes, SA-CPITR certifies terminal criticality on all five with a total of only three anchor events, while the corresponding solver-free variant certifies two. A sixth problem extends the method to state constraints, with certified feasibility and descent. The experiments use a fixed quadrature grid to evaluate integrals inside the optimization loop, and then independently re-verify certificates at every returned point by exact integration or continuum-valid enclosures.