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非常少(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，而不需要专家输入。这些方法提供了非常不同的方法来解决工业和科学中出现的难题。然而，在医疗保健、金融、物理等领域有一大类重要问题，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，而且获取数据的成本相对较高。传统的机器学习模型使用数据的效率太低，对这类问题毫无用处，数学模型很难捕捉到所有的依赖关系，而不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能学到的东西。我们试图扩大对这些模型及其数值的理解，使用应用程序作为研究的指南。我们简要概述了当前的两个项目。首先，摩根大通的尼古拉斯·维克托伊尔利用神经网络(一种强大的机器学习模型)为百慕大的掉期交易定价。百慕大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必须满足无套利的财务和数学约束，而传统的神经网络模型都无法满足这一点。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过程。我们没有尝试使用机器学习来直接获得百慕大掉期价格并遇到套利，而是使用网络来学习更好的校准程序初始化，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组的可逆解。在古代，微分方程一直是占主导地位的数学建模范例。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范围。神经微分方程表示具有非结构化向量场的微分方程，可以通过神经网络来适应手头的问题。这类模型已被证明是非常强大的，但优化过程依赖于数值方法，当与神经网络结合使用时，这些方法要么固有地不准确，要么非常耗费计算机内存。可逆数值方法起源于70年代的数值物理问题，但后来在神经微分方程式中被忽略。我们认为，实际上可逆方法既没有标准数值方法所遇到的问题，也没有进一步了解哪些数值方法是可逆的。在EPSRC的研究领域和战略网站上，这一主题属于“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."