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Random environments, stochastic equations, and randomized algorithms

随机环境、随机方程和随机算法

基本信息

批准号：
EP/V027824/1
负责人：
Benjamin Fehrman
金额：
$ 108.76万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Fellowship
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%2FV027824%2F1
关键词：
Random environments stochastic equations randomized

项目摘要

The overarching aim of my research is to understand and exploit randomness as it occurs in a diverse collection of settings. The techniques draw broadly from the mathematical discipline of stochastic analysis, which is based on the interplay of random processes, partial differential equations, and dynamical systems, to understand problems in (I) stochastic homogenization, (II) stochastic partial differential equations, and (III) machine learning.I. The fundamental observation of stochastic homogenization is that random phenomena can behave as though they are deterministic. For example, the conductance of a metal is significantly affected by the presence of microscopic impurities, which may arise from flaws in a manufacturing process or from environmental contamination. The complicated microstructure of these impurities makes them impossible to simulate efficiently with standard numerical methods. In stochastic homogenization, we instead identify a simple deterministic model that closely approximates the original material. We do this by establishing a complicated nonlinear averaging in what is effectively a random environment. The objectives of this proposal will use homogenization theory to characterize the properties of complex materials and turbulent fluids.II. While stochastic homogenization describes random phenomena that are effectively deterministic, some apparently deterministic phenomena are effectively random. This is the case when the stock market reacts to political events, or during the growth of a forest fire. The randomness is driven by essentially unquantifiable fluctuations at the microscopic level, like the reactions of individual investors or variations in the forest floor. We model these phenomena using equations driven by random noise, which are called stochastic partial differential equations (SPDEs). Because of the driving noise, SPDEs are not classically defined and making sense of their solutions is a difficult problem. The objectives of this proposal will develop a solution theory for classes of SPDEs that model rare events, like the extreme concentration of heat or energy in a mechanical system. The results will make rigorous long-standing informal connections between SPDEs and interacting particle systems.III. The goal of machine learning is to identify the essential features of large data sets and to create artificial neural networks with predictive power. For example, the development of image recognition and artificial intelligence technologies relies on the training of deep networks over an enormous amount of information. However, the scale of modern data makes the implementation of classical training techniques like gradient descent computationally infeasible. We overcome this problem by deliberately introducing randomness into the algorithm. Stochastic gradient descent (SGD) is a randomized process that optimizes at each step over a small but random sample of the data. SGD is the most common way to train neural networks, yet there is no rigorous justification for its convergence. The objectives of this proposal will develop a quantitative understanding of convergence for SGD and will characterize the loss landscape in deep learning.

我的研究的首要目标是理解和利用随机性，因为它发生在不同的设置集合。这些技术广泛地借鉴了随机分析的数学学科，该学科基于随机过程，偏微分方程和动力系统的相互作用，以理解（I）随机均匀化，（II）随机偏微分方程和（III）机器学习中的问题。随机均匀化的基本观察是随机现象可以表现得好像它们是确定性的。例如，金属的导电性受到微观杂质的显著影响，这些杂质可能来自制造过程中的缺陷或环境污染。这些杂质的复杂微观结构使得它们不可能用标准的数值方法有效地模拟。在随机均匀化中，我们确定了一个简单的确定性模型，它非常接近原始材料。我们通过在实际上是随机的环境中建立一个复杂的非线性平均来做到这一点。该提案的目标将使用均匀化理论来表征复杂材料和湍流流体的性质。虽然随机均匀化描述的随机现象实际上是确定的，但一些明显确定的现象实际上是随机的。当股市对政治事件作出反应时，或者在森林大火蔓延期间，情况就是如此。这种随机性是由微观层面上基本上无法量化的波动驱动的，比如个人投资者的反应或森林地面的变化。我们使用由随机噪声驱动的方程来模拟这些现象，这些方程被称为随机偏微分方程（SPDE）。由于驱动噪声，SPDE不是经典定义的，并且理解它们的解决方案是一个困难的问题。该提案的目标是为模拟罕见事件的SPDE类开发解决方案理论，例如机械系统中热量或能量的极端集中。结果将严格的长期存在的非正式联系之间的SPDE和相互作用的粒子系统。III。机器学习的目标是识别大型数据集的基本特征，并创建具有预测能力的人工神经网络。例如，图像识别和人工智能技术的发展依赖于在大量信息上对深度网络的训练。然而，现代数据的规模使得经典训练技术（如梯度下降）的实现在计算上不可行。我们通过故意在算法中引入随机性来克服这个问题。随机梯度下降（SGD）是一个随机化的过程，它在每一步都对一个小但随机的数据样本进行优化。SGD是训练神经网络最常见的方法，但它的收敛性没有严格的理由。该提案的目标是对SGD的收敛性进行定量理解，并描述深度学习中的损失情况。