Training Neural Networks to Discover Stochastic Differential Equation Based Models

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Differential equations are probably the most important tool on a mathematical modeller's workbench. The enormous variety of phenomena which have been successfully modelled using these types of equations is quite astounding - many of the fundamental laws of physics and chemistry are formulated as differential equations. Many complex systems in biology and economics too are modelled this way. In recent times, neural networks have become an increasingly powerful and popular method for creating models directly from data, without requiring the user to have an understanding of the underlying processes involved to build a powerful predictive model. These two modelling methods are now being researched together - the idea is to train neural networks to discover good differential equation-based models from data. However, differential equations have their limitations in what types of dynamics they can effectively model. Phenomenon such as financial markets and many biological processes are not well modelled using this technique as they display many sharp changes due to random perturbations which differential equations cannot express. A typical family of methods for modelling these phenomena, closely related to the differential equations discussed above, are stochastic differential equations. In the same way that neural networks can be trained on data to produce differential equation models, similarly, these stochastic models can be discovered by neural networks. However, some of the tricks that are used to make training these neural networks efficient for differential equations are not easy to transfer to the case of stochastic differential equations. In this project, we will be exploring what the most efficient and robust methods for training neural networks to discover stochastic differential equation-based models from data are. This has many applications including modelling financial markets and biological processes.