AONN：一种求解参数最优控制问题的伴随神经网络方法
AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems
摘要：
由偏微分方程（PDE）控制的参数最优控制问题在科学和工程应用中有着广泛的应用。传统的基于网格的数值方法求解此类问题通常需要对不同参数设置的偏微分方程进行重复求解，这在计算上是不允许的，特别是对于高维参数空间的问题。虽然最近提出的神经网络方法可以同时获得不同参数的最优解，但在处理具有复杂约束的问题时仍然存在挑战。在本文中，我们提出了AONN，伴随导向的神经网络方法，以克服现有的方法在解决参数最优控制问题的局限性。在AONN中，神经网络作为控制、伴随和状态函数的参数代理模型，一次性获得最优解。为了降低训练难度和处理复杂的约束条件，我们引入了一个迭代训练框架的灵感来自经典的直接伴随循环（DAL）方法，使惩罚项所产生的Karush-Kuhn-Tucker（KKT）系统可以避免。一旦训练完成，就可以通过神经网络的前向传播快速计算出参数特定的最优解，这可以进一步用于分析最优解的参数特性。一系列的数值实验表明，AONN的有效性和效率的问题，涉及各种类型的参数。
英文摘要：
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.
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