Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen-Cahn equation describes pattern formation associated to phase transitions. In finance, instead, the Black-Scholes equation describes the evolution of the price of derivative investment instruments. Such modern applications often require to solve these equations in high-dimensional regimes in which classical approaches are ineffective. Recently, an interesting new approach based on deep learning has been introduced by E, Han, and Jentzen [1][2]. The main idea is to construct a deep network which is trained from the samples of discrete stochastic differential equations underlying Kolmogorov's equation. The network is able to approximate, numerically at least, the solutions of the Kolmogorov equation with polynomial complexity in whole spatial domains. In this contribution we study variants of the deep networks by using different discretizations schemes of the stochastic differential equation. We compare the performance of the associated networks, on benchmarked examples, and show that, for some discretization schemes, improvements in the accuracy are possible without affecting the observed computational complexity.
翻译:非线性部分差异的科尔莫戈罗夫方程式被成功地用于描述自然科学、工程学甚至金融领域一系列广泛的时间依赖现象,例如,在物理系统中,艾伦-卡恩方程式描述与阶段过渡相关的模式形成。相反,在金融中,黑分方程式描述衍生投资工具价格的演变。这种现代应用往往需要在传统方法无效的高维系统中解决这些方程式。最近,E、Han和Jentzen采用了基于深层次学习的有趣新方法[1][2]。主要想法是建立一个深层次网络,从科尔莫戈罗夫方程式的离散分异差异方程式样本中加以培训。在数字上至少能够将科尔莫戈罗夫方程式的解决方案与整个空间域的多元复杂度相近。在这个贡献中,我们通过使用不同分解的分解公式来研究深层次网络的变量。我们根据基准示例比较了相关网络的运行情况,并表明,对于某些离心化方案而言,在不精确度上观察到的精确度是可能的改进。