Non-linear partial differential equations, stochastic representations, and numerical approximation by deep learning

非线性偏微分方程、随机表示和深度学习数值逼近

• 批准号: EP/W004070/1
• 负责人: Johannes Ruf
• 金额: $ 10.18万
• 依托单位: London School of Economics and Political Science
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW004070%2F1
• 关键词: Non; linear; partial; differential; equations

Geometric differential equations, describing how certain surfaces evolve, emerge in stochastic control problems, for example motivated in mathematical finance. The worst-case model for a long-term investor exploiting market volatility turns out to be characterised by such an equation. Well-established methods are used to solve differential equations for which analytically explicit solutions are not known. One approach is to use numerical methods (for example, finite difference and finite element schemes). An alternative is to reformulate such equations in probabilistic terms and to use simulation methods (for example, Monte Carlo and dynamic programming). The first approach performs very well in small dimensions but suffers from the curse of dimensionality. The second approach tends to be easily implementable but is usually slow since each simulation provides the solution at only one point.In the recent few years new exciting research has been proposed (several researchers working on this field in and outside the UK are named below) to 'learn' the solution. To this end, the solution is approximated by a (shallow or deep) neural network, i.e., through a composition of several linear and non-linear functions. The justification of this method is an application of the so-called universal approximation results as a core idea. The neural network is 'trained' by standard backpropagation techniques. There are different proposals for the corresponding functional to be minimised; e.g. based on the minimisation of certain operator norms in the context of adversarial learning or using stochastic representations (e.g. via Feynman-Kac). The later approach, for example, has been very successfully applied to differential equations that can be associated to backward stochastic differential equations.For this project the PI plans to work on developing this approach to a large class of geometric differential equations. Due to their importance (e.g. in physics) these equations have been studied extensively in differential geometry and in the partial differential equations' literature. One important example is the mean curvature flow, which is described by these equations. The seminal work by Soner and Touzi connects these equations to stochastic control problems. Similarly, Kohn and Serfaty link them to the value functions of deterministic games between two players. In a similar spirit as Soner and Touzi, the PI and Larsson have recently established a relationship between a certain class of these differential equations (especially those ones de- scribing the so-called 'minimum curvature' flow) to a specific control problem that describes the time a martingale can be contained in a compact set.These links between geometric differential equations and stochastic control problems (and also deterministic games) open up a promising avenue to find good numerical approximations to such high-dimensional problems. The PI plans to exploit these links to design and study the use of neural networks as a numerical approximation to geometric differential equations, starting with the minimum curvature flow.The project has an experimental component that implements the algorithms and provides insights in the speed of convergence, the necessary number of layers, learning rates, other hyperparameters, and the effect of exploiting symmetries (in the domain of the differential equation). This component also yields a proof-of-concept and will be made publicly availably via a GitHub repository. The second component of the project establishes expression rates and additional properties rigorously.

几何微分方程描述了某些表面如何发展，在随机控制问题中出现，例如在数学金融中动机。长期投资者利用市场波动的最糟糕的模型原来是这种方程式的特征。建立良好的方法用于求解尚不清楚分析明确解的微分方程。一种方法是使用数值方法（例如，有限差和有限元方案）。一种替代方法是用概率术语重新重新制定此类方程，并使用仿真方法（例如，蒙特卡洛和动态编程）。第一种方法在小维度上表现良好，但遭受了维度的诅咒。第二种方法往往很容易实现，但通常很慢，因为每个模拟仅提供了一个点。在最近几年中，提出了新的激动人心的研究（在下面命名了英国和外部的一些研究人员）来“学习”解决方案。为此，解决方案是通过（浅或深）神经网络近似的，即通过几种线性和非线性函数的组成。该方法的理由是所谓的通用近似结果作为核心思想的应用。神经网络是通过标准反向传播技术“训练”的。对于相应的功能有所最小化，有不同的建议。例如基于在对抗性学习或使用随机表示的背景下某些操作员规范的最小化（例如，通过feynman-kac）。例如，以后的方法已非常成功地应用于可以与后退随机微分方程相关联的微分方程。对于此项目，PI计划致力于将此方法开发到大量的几何微分方程。由于它们的重要性（例如在物理学中），这些方程在差异几何形状和部分微分方程文献中进行了广泛研究。一个重要的例子是平均曲率流，这些曲率流由这些方程式描述。 Soner和Touzi的开创性工作将这些方程式连接到随机控制问题。同样，Kohn和Sefaty将它们与两个玩家之间确定性游戏的价值功能联系起来。 PI和Larsson以与Soner和Touzi相似的精神，最近建立了某些类别的差分方程之间的关系（尤其是那些将所谓的“最小曲率”流动到特定的控制问题）与特定的控制问题的关系，这些问题描述了时间的时间，这些时间可以包含在紧凑的集合中。对于此类高维问题的良好数值近似值。 PI计划利用这些链接，以设计和研究神经网络作为对几何微分方程的数值近似，从最小曲率流开始。该项目具有实验组件，可以实现算法，并提供实现速度的速度，所需的层次，其他超级范围的范围，效应效应效应的效果，并实现速度的速度，并提供差异。等式）。该组件还产生概念验证，并将通过GitHub存储库公开提供。该项目的第二个组成部分严格建立了表达率和其他属性。

项目成果

Minimum curvature flow and martingale exit times

最小曲率流量和鞅退出时间

• 发表时间: 2022
• 作者: Larsson M