论文标题
一阶HJB方程的神经网络和用障碍术语应用到前繁殖的应用
Neural networks for first order HJB equations and application to front propagation with obstacle terms
论文作者
论文摘要
我们考虑一个确定性的最佳控制问题，在有限的地平线上具有最大运行成本功能，并为贝尔曼的动态编程原理提出了深层的神经网络近似，也对应于一些一阶汉密尔顿 - 雅各布·贝尔曼方程。这项工作遵循Huré等人的界限。 （Siam J.Numer。Anal。，第59（1）卷，2021年，第525-557页），其中在随机背景下提出了算法。但是，我们需要开发一种全新的方法，以应对确定性环境中错误的传播，在确定性设置中，动态中不存在扩散。我们的分析给出了平均规范的精确误差估计。然后，在存在障碍物约束的情况下，在与前繁殖模型有关的几个学术数值示例中进行了说明，显示了该方法与平均维度的相关性（例如，从$ 2 $到$ 8 $），即使在非平滑价值功能方面也是如此。
We consider a deterministic optimal control problem with a maximum running cost functional, in a finite horizon context, and propose deep neural network approximations for Bellman's dynamic programming principle, corresponding also to some first-order Hamilton-Jacobi-Bellman equations. This work follows the lines of Huré et al. (SIAM J. Numer. Anal., vol. 59 (1), 2021, pp. 525-557) where algorithms are proposed in a stochastic context. However, we need to develop a completely new approach in order to deal with the propagation of errors in the deterministic setting, where no diffusion is present in the dynamics. Our analysis gives precise error estimates in an average norm. The study is then illustrated on several academic numerical examples related to front propagations models in the presence of obstacle constraints, showing the relevance of the approach for average dimensions (e.g. from $2$ to $8$), even for non-smooth value functions.
