Abstract

Chaos theory is a branch of modern mathematics concerning the non-linear dynamic systems that are highly sensitive to their initial states. It has extensive real-world applications, such as weather forecasting and stock market prediction. The Lorenz system, defined by three ordinary differential equations (ODEs), is one of the simplest and most popular chaotic models. Historically research has focused on understanding the Lorenz system's mathematical characteristics and dynamical evolution including the inherent chaotic features it possesses. In this thesis, we take a data-driven approach and propose the task of predicting future states of the chaotic system from limited observations. We explore two directions, answering two distinct fundamental questions of the system based on how informed we are about the underlying model. When we know the data is generated by the Lorenz System with unknown parameters, our task becomes parameter estimation (a white-box problem), or the ``inverse'' problem. When we know nothing about the underlying model (a black-box problem), our task becomes sequence prediction. We propose two algorithms for the white-box problem: Markov-Chain-Monte-Carlo (MCMC) and a Multi-Layer-Perceptron (MLP). Specially, we propose to use the Metropolis-Hastings (MH) algorithm with an additional random walk to avoid the sampler being trapped into local energy wells. The MH algorithm achieves moderate success in predicting the $\rho$ value from the data, but fails at the other two parameters. Our simple MLP model is able to attain high accuracy in terms of the $l_2$ distance between the prediction and ground truth for $\rho$ as well, but also fails to converge satisfactorily for the remaining parameters. We use a Recurrent Neural Network (RNN) to tackle the black-box problem. We implement and experiment with several RNN architectures including Elman RNN, LSTM, and GRU and demonstrate the relative strengths and weaknesses of each of these methods. Our results demonstrate the promising role of machine learning and modern statistical data science methods in the study of chaotic dynamic systems. The code for all of our experiments can be found on \url{https://github.com/Yajing-Zhao/}

Degree

College and Department

Physical and Mathematical Sciences; Mathematics

Rights

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