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Parametric optimal control problems governed by partial differentialequations (PDEs) are widely found in scientific and engineering applications.Traditional grid-based numerical methods for such problems generally requirerepeated solutions of PDEs with different parameter settings, which iscomputationally prohibitive especially for problems with high-dimensionalparameter spaces. Although recently proposed neural network methods make itpossible to obtain the optimal solutions simultaneously for differentparameters, challenges still remain when dealing with problems with complexconstraints. In this paper, we propose AONN, an adjoint-oriented neural networkmethod, to overcome the limitations of existing approaches in solvingparametric optimal control problems. In AONN, the neural networks are served asparametric surrogate models for the control, adjoint and state functions to getthe optimal solutions all at once. In order to reduce the training difficultyand handle complex constraints, we introduce an iterative training frameworkinspired by the classical direct-adjoint looping (DAL) method so that penaltyterms arising from the Karush-Kuhn-Tucker (KKT) system can be avoided. Once thetraining is done, parameter-specific optimal solutions can be quickly computedthrough the forward propagation of the neural networks, which may be furtherused for analyzing the parametric properties of the optimal solutions. Thevalidity and efficiency of AONN is demonstrated through a series of numericalexperiments with problems involving various types of parameters.

AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems