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:
- 金额:$ 10.18万
- 依托单位:
- 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:
- 关键词:
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,已经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) “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)与一个特定的控制问题之间建立了一种关系,该控制问题描述了鞅可以包含在紧集合中的时间。几何微分方程和随机控制问题(以及确定性博弈)之间的这些联系为找到此类高维问题的良好数值近似开辟了一条有希望的途径。 PI 计划利用这些链接来设计和研究神经网络作为几何微分方程数值逼近的用途,从最小曲率流开始。该项目有一个实验组件,可以实现算法并提供收敛速度的见解、必要的层数、学习率、其他超参数以及利用对称性的效果(在微分方程领域)。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表达率和附加特性。
项目成果
期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Minimum curvature flow and martingale exit times
最小曲率流量和鞅退出时间
- DOI:
- 发表时间:2022
- 期刊:
- 影响因子:0
- 作者:Larsson M
- 通讯作者:Larsson M
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Johannes Ruf其他文献
Convergence in models with bounded expected relative hazard rates
具有有限预期相对危险率的模型的收敛
- DOI:
10.1016/j.jet.2014.09.014 - 发表时间:
2012 - 期刊:
- 影响因子:0
- 作者:
Carlos Oyarzún;Johannes Ruf - 通讯作者:
Johannes Ruf
Energy Hub Gas: A Modular Setup for the Evaluation of Local Flexibility and Renewable Energy Carriers Provision
能源中心天然气:用于评估当地灵活性和可再生能源载体供应的模块化设置
- DOI:
10.1109/sege55279.2022.9889751 - 发表时间:
2022 - 期刊:
- 影响因子:0
- 作者:
Rafael Poppenborg;Malte Chlosta;Johannes Ruf;C. Hotz;Clemens Düpmeier;T. Kolb;V. Hagenmeyer - 通讯作者:
V. Hagenmeyer
NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS
使用基于鞅矩的混合危险模型的非参数识别
- DOI:
- 发表时间:
2019 - 期刊:
- 影响因子:0.8
- 作者:
Johannes Ruf;James Lewis Wolter - 通讯作者:
James Lewis Wolter
Auxiliary Power Supply for a Semiconductor-based Marx Generator
基于半导体的马克思发生器的辅助电源
- DOI:
10.1109/optim-acemp50812.2021.9590076 - 发表时间:
2021 - 期刊:
- 影响因子:0
- 作者:
M. Sack;Johannes Ruf;D. Herzog;G. Mueller - 通讯作者:
G. Mueller
Weak Tail Conditions for Local Martingales
本地鞅的弱尾条件
- DOI:
- 发表时间:
2015 - 期刊:
- 影响因子:0
- 作者:
H. Hulley;Johannes Ruf - 通讯作者:
Johannes Ruf
Johannes Ruf的其他文献
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{{ truncateString('Johannes Ruf', 18)}}的其他基金
Predictable Variations in Stochastic Calculus
随机微积分的可预测变化
- 批准号:
EP/Y024524/1 - 财政年份:2023
- 资助金额:
$ 10.18万 - 项目类别:
Research Grant
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