On the Interaction of Machine Learning, Stochastic Analysis, and Applications

传统的数学建模和随机分析使用领域的专家知识来解决数据很少（如果有的话）的问题。机器学习是人工智能的一个子集，它使用大量数据来学习给定问题中的复杂依赖关系，而无需专家输入。这些方法提供了解决工业和科学中出现的困难问题的非常不同的方法。然而，在医疗保健、金融、物理学等领域有很多重要的问题，这两种方法都不能令人满意。这通常是因为这些学科中的问题极其复杂，并且获取数据相对昂贵。传统的机器学习模型对数据的使用效率太低，无法在此类问题上发挥作用，而数学模型则难以捕捉所有的依赖关系，无法使用任何可用的数据。科学机器学习的现代方法融合了这两个学科，利用数学模型的已知结构，并使用机器学习从数据中学习我们所能做的。我们寻求扩大对这些模型及其数值的理解，使用应用程序作为研究指南。我们简要介绍目前的两个项目。第一个是与JPMorgan的Nicolas Victoir合作，利用神经网络（一种功能强大的机器学习模型）为丹麦掉期定价。丹麦掉期是固定收益衍生品市场上流行的复杂金融工具。这些掉期交易的价格需要满足无套利的金融和数学约束，这是传统神经网络模型无法满足的。然而，金融数学中的传统随机微分方程模型确实满足无套利约束，代价是需要定期进行昂贵的校准程序。我们没有尝试使用机器学习来直接获得可转换价格并遇到套利，而是使用网络来学习校准过程的更好初始化，从而大大降低了执行校准的成本。第二个项目是关于神经微分方程的可逆解算器。微分方程一直是古代占主导地位的数学建模范式。在过去的一百年里，数值方法的出现进一步拓宽了微分方程的应用。神经微分方程表示具有非结构化向量场的微分方程，这些向量场可以通过神经网络适应手头的问题。这种模型已被证明是非常强大的，但优化过程依赖于数值方法，这是固有的不准确或非常计算机内存密集型时，与神经网络一起使用。可逆数值方法在70年代从数值物理中的问题中出现，但随后在神经微分方程的背景下被忽略。我们认为，事实上可逆的方法遭受的标准数值方法所遇到的问题，也进一步了解正是哪些数值方法可以可逆的。在EPSRC网站的研究领域和战略，这福尔斯属于该地区的“数学科学主题。"

Traditional mathematical modelling and stochastic analysis use expert knowledge of a domain to solve a problem with very little (if any) data. Machine learning is a subset of artificial intelligence which uses large amounts of data to learn the complex dependencies in a given problem with no expert input. These approaches offer very different ways of solving difficult problems which arise in industry and the sciences. However, there is a large class of important problems in healthcare, finance, physics, etc. for which neither approach is satisfactory. This is often because problems in these disciplines are extremely complex, and data is relatively expensive to acquire. Traditional machine learning models use data too inefficiently to be useful on such problems, and mathematical models struggle to capture all of the dependencies, using none of what data is available. Modern methods in scientific machine learning blend these two disciplines, utilising the known structure of mathematical models and learning what we can from data using machine learning. We seek to expand the understanding of these models and their numerics, using applications as a guide for research. We briefly outline two current projects. The first, with Nicolas Victoir of JPMorgan, is that of pricing Bermudan swaptions using neural networks (a powerful machine learning model). Bermudan swaptions are complex financial instruments popular in fixed income derivatives markets. Prices for these swaptions are required to satisfy the financial and mathematical constraint of no-arbitrage, which no traditional neural network model satisfies. However, traditional stochastic differential equation models in financial mathematics do satisfy the no-arbitrage constraint, at the cost of needing to regularly undergo a costly calibration procedure. Rather than try and use machine learning to directly get the Bermudan swaption prices and encounter arbitrage, we instead use the network to learn better initializations for the calibration procedure, drastically reducing the cost of performing calibration. The second project is on reversible solvers for neural differential equations. Differential equations have been the dominant mathematical modelling paradigm to antiquity. In the last hundred years, numerical methods have emerged which have broadened the application of differential equations further still. Neural differential equations represent differential equations with unstructured vector fields that can be adapted to the problem at hand via neural networks. Such models have proven to be very powerful, but the optimisation procedure relies on numerical methods which are either inherently inaccurate or very computer memory intensive when used in conjunction with neural networks. Reversible numerical methods arose in the 70s from problems in numerical physics, but have subsequently been ignored in the context of neural differential equations. We argue that in fact reversible methods suffer from neither of the problems encountered by standard numerical methods, and also further the understanding of precisely which numerical methods can be made reversible. On the EPSRC website of research areas and strategies, this falls under the area of "Mathematical Sciences Theme."