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。可以通过产生确定性混沌的数学模型来研究它。混沌系统对初始条件很敏感，这使得它们很难预测。灵敏度是用李雅普诺夫指数来量化的，在大多数情况下，必须使用数值技术来求出该指数。最近有人建议在这种系统中使用深度学习人工神经网络(ANN)进行预测，深度学习算法可以被训练来预测混沌系统的未来行为[1，2]。还可以教他们辨别混乱的行为。预测和识别也可以与分形学一起使用，看看人工神经网络是否能够继续/复制分形图，或者从输入图像或时间序列数据中识别分形图。该项目的目标是通过研究分形图人工智能(AI)[3]和相关算法识别混沌或分形图的能力，进一步扩展这一方向。我还想考虑[3]中列出的一些(AI)研究主题--分布式计算、学习能力和实时决策。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，我是在过去两年里为我的FYP和数学项目工作而获得的。将根据项目的方向考虑不同的编程语言。使用Python将是一种很好的语言，但它需要一些时间来学习。另一种选择是使用MatLab，因为较新的版本提供了研究深度学习的工具，我已经有近4年的程序经验。根据任务的需要，也可以使用其他语言。Draft计划从研究现有的机器学习算法开始，重点是深度学习(ANN)和分形人工智能。稍后，研究将转向预测和/或更具体的研究主题(AI)的进一步工作。一个受限的三体问题最初将被用来作为该方法的基准，但在适当的时候将扩大到包括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.