An adaptive adjoint-oriented neural network for solving parametric optimal control problems with singularities

In this work, we present an adaptive adjoint-oriented neural network (adaptive AONN) for solving parametric optimal control problems governed by partial differential equations. The proposed method integrates deep adaptive sampling techniques with the adjoint-oriented neural network (AONN) framework. It alleviates the limitations of AONN in handling low-regularity solutions and enhances the generalizability of deep adaptive sampling for surrogate modeling without labeled data (${\text{DAS}}^2$). The effectiveness of the adaptive AONN is demonstrated through numerical examples involving singularities.