论文标题
基于近似定向衍生物的最佳控制PDE的最佳控制算法
A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives
论文作者
论文摘要
我们提出和分析了一种数值算法，用于解决学习信息的半连线部分偏微分方程的一类最佳控制问题。后者是一类PDE，其成分原则上未知，并由非滑动relu神经网络近似。我们首先表明，RELU网络的直接平滑目的是使用经典的数值求解器可能具有某些缺点，即可能引入相应状态方程的多个解决方案。这促使我们设计了一种数值算法，该算法通过采用受无束方法启发的下降算法直接处理非平滑最佳控制问题。提供了几个数值示例，并显示了算法的效率。
We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations. The latter is a class of PDEs with constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that a direct smoothing of the ReLU network with the aim to make use of classical numerical solvers can have certain disadvantages, namely potentially introducing multiple solutions for the corresponding state equation. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown.
