Enhanced physics-informed neural networks for PDE-constrained optimal control: A synergistic approach with adversarial attack and scale adjustment

Optimal control problems constrained by partial differential equations (PDEs) are widely studied in science and engineering, with physics-informed neural networks emerging as a powerful tool for solving such problems. However, existing methods often encounter difficulties when dealing with complex problem structures and show sensitivity to varying regularization parameters. To address these challenges, we propose ASNN, an adversarial and scale-adjusted neural network. ASNN is formulated within the Karush–Kuhn–Tucker framework and utilizes two independent neural networks to approximate the state, adjoint, and control variables. By incorporating adversarial attacks, ASNN enhances the accuracy of the solution, while the scale adjustment strategy improves numerical stability under different regularization settings. Extensive numerical experiments on benchmark PDE-constrained optimal control problems, including those governed by the Poisson’s equation, semilinear equation, control-constrained equation, and Navier–Stokes and Burgers’ equations, demonstrate the effectiveness and robustness of the proposed method and highlight its advantages over existing approaches.