Use of machine learning to forecast the behaviour of chaotic dynamical systems

研究项目将涉及数学建模、计算机模拟和机器学习。最初的想法是使用机器学习来预测混沌动力系统的行为。混沌动力学出现在许多自然系统中，从物理和生物到天气和气候应用。它可以通过产生确定性混沌的数学模型来研究。混沌系统对初始条件非常敏感，这使得混沌系统难以预测。灵敏度是用李亚普诺夫指数来量化的，在大多数情况下必须使用数值技术。最近有人建议[1,2]在这类系统中使用深度学习人工神经网络（ANN）进行预测。可以训练深度学习算法来预测混沌系统的未来行为[1,2]。它们还可以学会识别混沌行为。预测和识别也可以与分形一起使用，看看人工神经网络是否可以继续/复制分形，或者从输入图像或时间序列数据中识别分形模式。该项目的目标是通过研究分形人工智能（AI）[3]和相关算法来进一步扩展这一方向，以识别混沌或分形模式。我还想考虑[3]中列出的一些（人工智能）研究课题——分布式计算、学习能力和实时决策。关于意识或普遍性的话题也很有趣。方法论和编程语言：数学部分将建立在我现有的教育背景和混沌动力系统知识的基础上，我在过去两年的五年计划和MMath项目中获得了这些知识。根据项目的发展方向，可以考虑使用不同的编程语言。Python是一种很好用的语言，但是需要一些时间来学习。另一种选择是使用MATLAB作为新版本提供的工具来研究深度学习，我有近4年的程序经验。也可以根据任务的需要使用其他语言。计划草案将从研究现有的机器学习算法开始，重点研究深度学习（ANN）和分形人工智能。后来，研究将转向预测和/或更具体的研究主题（人工智能）的进一步工作。最初将使用一个受限的三体问题作为方法的基准，但这将在适当的时候扩展，以纳入其他类型的动力系统。

Research project would involve mathematical modelling, computer simulations and machine learning. Initial idea is to use machine learning to forecast the behaviour of chaotic dynamical systems.Chaotic dynamic appears in many natural systems, ranging from physical and biological to weather and climate applications. It can be studied by mathematical models that generate deterministic chaos. Chaotic systems are sensitive to the initial conditions, which makes them difficult to predict. Sensitivity is quantified by Lyapunov exponents that in most cases must be found using numerical techniques. Recently it was suggested [1,2] to use deep learning artificial neural networks (ANN) for forecasting in such systems.Deep learning algorithms can be trained to predict the future behaviour of chaotic systems [1,2]. They can also be taught to recognise chaotic behaviour [4]. Prediction and recognition could also be used with fractals to see if ANN can continue/reproduce fractals or recognise fractal patterns from input image or time-series data.The aim of the project would be to extend such a direction further by studying fractal artificial intelligence (AI) [3] and related algorithms ability to recognise chaos or fractal patterns. I would also like to consider some of the (AI) research topics listed in [3] - distributed computing, learning capabilities and real-time decision-making. Topics of consciousness or universality are also intriguing.Methodology and Programming Languages: The mathematical component will be built upon my existing educational background and knowledge of chaotic dynamical systems I have acquired in the last two years working on my FYP and MMath projects. Different programming languages would be considered based on the direction that the project takes. Python would be a good language to use, but it would take some time to learn. Another option is to use MATLAB as newer versions provide tools to study deep learning, and I have almost 4 years of experience with the program. Other languages can also be used, as appropriated for the task.Draft plan is to start by studying the existing machine learning algorithms, focusing on deep learning (ANN) and fractal AI. Later the research would shift to forecasting and/or further work on more specific research topics (AI). A Restricted three-body problem will be initially used to benchmark the methodology, but this will be expanded in due course to incorporate other types of dynamical systems as well.