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

描述某些表面如何演化的几何微分方程式出现在随机控制问题中，例如在数学金融中。对于利用市场波动性的长期投资者来说，最糟糕的情况是这样一个等式。已有的方法被用来求解解析显式解未知的微分方程式。一种方法是使用数值方法(例如，有限差分和有限元格式)。另一种选择是用概率术语重新表述这些方程，并使用模拟方法(例如，蒙特卡洛和动态规划)。第一种方法在小维度上表现得很好，但受到维度诅咒的影响。第二种方法往往很容易实现，但通常很慢，因为每个模拟只在一个点上提供解决方案。最近几年，提出了新的令人兴奋的研究(在英国国内外从事这一领域的几名研究人员如下所述)来“学习”解决方案。为此，通过(浅或深)神经网络，即通过几个线性和非线性函数的组合来逼近解。这种方法的合理性是以所谓的普适近似结果为核心思想的应用。神经网络是通过标准的反向传播技术进行“训练”的。对于要最小化相应的泛函，存在不同的建议；例如，基于对抗性学习环境中的某些算子范数的最小化，或者使用随机表示(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关联到倒向随机微分方程的微分方程。在这个项目中，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组的文献中得到了广泛的研究。一个重要的例子是由这些方程描述的平均曲率流。索纳和图兹的开创性工作将这些方程与随机控制问题联系起来。同样，科恩和瑟法蒂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。与Siner和Touzi类似，PI和Larsson最近建立了一类此类微分方程(特别是那些描述所谓的最小曲率流)与描述一个特殊控制问题之间的关系，该控制问题描述了一个鞅可以包含在一个紧集内的时间。这些几何微分方程和随机控制问题(以及确定性对策)之间的联系为寻找此类高维问题的良好数值逼近开辟了一条很有希望的途径。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式的数值逼近，从最小曲率流开始。该项目有一个实验组件，它实现了算法，并提供了关于收敛速度、必要的层数、学习率、其他超参数以及利用对称性的效果(在微分方程式领域)的见解。该组件还提供了概念验证，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了表达速率和附加属性。

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.