A robust one-shot method based on adjoint surrogate model for PDE-constrained optimization

For PDE-constrained optimization, direct adjoint looping (DAL) solves the state and adjoint equations separately until convergence for per update of the design variables, providing accurate gradient and robust optimization. One-shot method simultaneously solves the state equation, adjoint equation, and design equation in a fully coupled system, requiring just O(1) forward PDE solves and significantly reducing simulation cost. However, one-shot systems often exhibit high condition numbers, which can lead to slow convergence or numerical divergence. This study develops a robust one-shot optimization framework based on adjoint surrogate model, where the surrogate model approximates the adjoint variables and is embedded into the coupled optimization process. Numerical experiments on three PDE-constrained benchmarks, including parameter identification and aerodynamic shape optimization, demonstrate that the present method achieves over an order-of-magnitude speedup compared with DAL. This study highlights the effects of using adjoint-system surrogates on the efficiency and robustness of one-shot optimization, providing a general and practical pathway for accelerating PDE-constrained design problems.