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

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."