Training Neural Networks to Discover Stochastic Differential Equation Based Models

在数学建模人员的工作台上，微分方程可能是最重要的工具。用这些类型的方程成功地模拟了各种各样的现象，这是相当令人震惊的--许多物理和化学的基本定律都是用微分方程式表示的。生物学和经济学中的许多复杂系统也是以这种方式建模的。近年来，神经网络已经成为一种日益强大和流行的方法，用于直接从数据创建模型，而不需要用户了解建立强大的预测模型所涉及的基本过程。这两种建模方法现在正在一起研究--其想法是训练神经网络，从数据中发现良好的基于微分方程的模型。然而，在可以有效建模的动力学类型方面，微分方程式有其局限性。金融市场和许多生物过程等现象不能很好地用这种技术建模，因为它们由于随机扰动而表现出许多急剧的变化，而这些随机扰动是微分方程式无法表达的。与上面讨论的微分方程密切相关的对这些现象进行建模的一类典型方法是随机微分方程。与神经网络可以对数据进行训练以产生微分方程模型的方式相同，这些随机模型也可以被神经网络发现。然而，用于使这些神经网络有效地用于微分方程式的一些技巧并不容易转移到随机微分方程式的情况。在这个项目中，我们将探索训练神经网络从数据中发现基于随机微分方程的模型的最有效和最健壮的方法是什么。它有许多应用，包括对金融市场和生物过程进行建模。

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.