  Parametric optimal control problems governed by partial differential
equations (PDEs) are widely found in scientific and engineering applications.
Traditional grid-based numerical methods for such problems generally require
repeated solutions of PDEs with different parameter settings, which is
computationally prohibitive especially for problems with high-dimensional
parameter spaces. Although recently proposed neural network methods make it
possible to obtain the optimal solutions simultaneously for different
parameters, challenges still remain when dealing with problems with complex
constraints. In this paper, we propose AONN, an adjoint-oriented neural network
method, to overcome the limitations of existing approaches in solving
parametric optimal control problems. In AONN, the neural networks are served as
parametric surrogate models for the control, adjoint and state functions to get
the optimal solutions all at once. In order to reduce the training difficulty
and handle complex constraints, we introduce an iterative training framework
inspired by the classical direct-adjoint looping (DAL) method so that penalty
terms arising from the Karush-Kuhn-Tucker (KKT) system can be avoided. Once the
training is done, parameter-specific optimal solutions can be quickly computed
through the forward propagation of the neural networks, which may be further
used for analyzing the parametric properties of the optimal solutions. The
validity and efficiency of AONN is demonstrated through a series of numerical
experiments with problems involving various types of parameters.
