Ever since the concepts of dynamic programming were introduced, one of the most difficult challenges has been to adequately address high-dimensional control problems. With growing dimensionality, the utilisation of Deep Neural Networks promises to circumvent the issue of an otherwise exponentially increasing complexity. The paper specifically investigates the sampling issues the Deep Galerkin Method is subjected to. It proposes a drift relaxation-based sampling approach to alleviate the symptoms of high-variance policy approximations. This is validated on mean-field control problems; namely, the variations of the opinion dynamics presented by the Sznajd and the Hegselmann-Krause model. The resulting policies induce a significant cost reduction over manually optimised control functions and show improvements on the Linear-Quadratic Regulator problem over the Deep FBSDE approach.

翻译：自动态规划理论提出以来，如何有效处理高维控制问题始终是重大挑战。随着问题维度的增长，深度神经网络的应用有望规避复杂度指数级上升的困境。本文重点研究深度伽辽金方法面临的采样问题，提出基于漂移松弛的采样策略以缓解高方差策略逼近的缺陷。该方法在平均场控制问题上得到验证，具体以Sznajd模型和Hegselmann-Krause模型的观点动力学变体为研究对象。实验表明，所得控制策略相较于人工优化控制函数能实现显著成本降低，并在线性二次调节器问题上较深度FBSDE方法表现出性能提升。