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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.

几何微分方程，描述了某些表面如何演变，出现在随机控制问题中，例如数学金融。对于一个长期投资者来说，利用市场波动的最坏情况模型就是这样一个方程。已建立的方法被用来解决微分方程的解析显式解决方案是未知的。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这些方程，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得很好，但受到维度灾难的影响。第二种方法易于实现，但通常很慢，因为每个模拟只在一个点上提供解决方案。在最近几年，新的令人兴奋的研究已经提出（几个研究人员在英国和英国以外的工作在这个领域被命名如下），以“学习”的解决方案。为此，通过（浅或深）神经网络来近似该解，即，通过几个线性和非线性函数的组合。这种方法的理由是所谓的通用近似结果作为核心思想的应用。神经网络通过标准的反向传播技术进行“训练”。对于相应的泛函有不同的最小化建议;例如，基于对抗学习背景下某些算子范数的最小化或使用随机表示（例如通过Feynman-Kac）。例如，后一种方法已经非常成功地应用于可以与向后随机微分方程相关联的微分方程。对于这个项目，PI计划将这种方法发展到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的文献中得到了广泛的研究。一个重要的例子是平均曲率流，这是由这些方程描述。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。类似地，Kohn和Serfaty将它们与两个参与者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最近建立了一类微分方程之间的关系，（特别是那些描述所谓的“最小曲率”流的问题）到一个特定的控制问题，描述了鞅可以包含在一个紧集中的时间。（以及确定性游戏）开辟了一个有前途的途径，找到良好的数值逼近这样的高维问题。PI计划利用这些联系来设计和研究使用神经网络作为几何微分方程的数值近似，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，必要的层数，学习率，其他超参数，以及利用对称性的效果（在微分方程的域中）。该组件还产生了概念验证，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地建立表达率和附加属性。