Physics-informed adjoint neural network for parametric PDE-constrained optimal control problems.

Many engineering applications involve solving parametric optimal control problems governed by partial differential equations (PDEs). Conventional numerical approaches suffer from high computational costs in multi-parameter scenarios, as they require solving the PDE system repeatedly for each parameter configuration, and remeshing may be also needed if domains vary with parameters. While deep learning offers a promising alternative by handling multiple parameters simultaneously, existing neural networks (NNs) still face challenges in accuracy and computational complexity. We propose two novel physics-informed adjoint NN (PIANN) methods for solving parametric optimal control problems. The PIANN1 employs a unified NN architecture that simultaneously approximates state and adjoint functions, optimized via optimality-condition loss minimization. The control solution is derived analytically from the adjoint-enhanced PDE system. The PIANN2 implements a coupled dual-network framework, where separate networks learn state and adjoint variables through an iterative direct-adjoint training process that inherently preserves constraints without penalty terms from Karush-Kuhn-Tucker (KKT) conditions. Both the PIANN1 and 2 simplify the network structure in comparison with the conventional NNs, and the PIANN2 avoids the involvement of penalty parameters for the KKT condition. The performance of these methods is verified by many benchmark parametric optimal control problems posed with the control constraint parameters, geometric parameters, corner singularities, and 3D domains. Numerical experiments show that the proposed methods achieve significantly higher accuracy compared to the existing NN techniques. In the appendix we further provide a theoretical analysis of the proposed methods, establishing residual-to-solution stability, convergence of the PIANN2 inexact direct-adjoint iteration under a small-gain condition, and a generalization bound that relates the empirical KKT loss to the continuous residual with explicit sample complexity. These results establish that the empirical residual minimization is a mathematically principled approach to solving parametric PDE-constrained optimal control problems.