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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

基本信息

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

项目摘要

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.

描述某些表面如何演化的几何微分方程出现在随机控制问题中，例如数学金融中的动机。对于利用市场波动的长期投资者来说，最坏情况模型的特征就是这样一个方程。成熟的方法用于求解未知解析显式解的微分方程。一种方法是使用数值方法（例如，有限差分和有限元方案）。另一种方法是用概率术语重新表述此类方程并使用模拟方法（例如蒙特卡罗和动态规划）。第一种方法在小维度上表现良好，但受到维度灾难的影响。第二种方法往往很容易实现，但通常很慢，因为每次模拟只提供一个点的解决方案。近年来，人们提出了新的令人兴奋的研究（下面列出了英国国内外从事这一领域工作的几位研究人员）来“学习”解决方案。为此，解决方案由（浅层或深层）神经网络近似，即通过几个线性和非线性函数的组合。该方法的合理性在于应用了所谓的万能逼近结果作为核心思想。神经网络通过标准反向传播技术进行“训练”。对于相应功能的最小化有不同的建议；例如基于对抗性学习背景下某些算子规范的最小化或使用随机表示（例如通过 Feynman-Kac）。例如，后一种方法已非常成功地应用于可与后向随机微分方程相关的微分方程。对于该项目，PI 计划致力于开发这种方法来处理一大类几何微分方程。由于它们的重要性（例如在物理学中），这些方程在微分几何和偏微分方程文献中得到了广泛的研究。一个重要的例子是平均曲率流，它由这些方程描述。 Soner 和 Touzi 的开创性工作将这些方程与随机控制问题联系起来。同样，Kohn 和 Serfaty 将它们与两个玩家之间的确定性博弈的价值函数联系起来。本着与 Soner 和 Touzi 相似的精神，PI 和 Larsson 最近在这些微分方程的某一类（特别是那些描述所谓“最小曲率”流的微分方程）与描述鞅可以包含在紧致集合中的时间的特定控制问题之间建立了关系。几何微分方程和随机控制问题（以及确定性博弈）之间的这些联系开辟了一条有希望的途径来寻找对此类高维问题的良好数值近似。 PI 计划利用这些链接来设计和研究使用神经网络作为几何微分方程的数值逼近，从最小曲率流开始。该项目有一个实验组件，用于实现算法并提供收敛速度、必要层数、学习率、其他超参数以及利用对称性的效果（在微分域中）的见解。方程）。该组件还产生概念验证，并将通过 GitHub 存储库公开提供。该项目的第二部分严格建立表达率和附加特性。