Characterizing information loss in a chaotic double pendulum with the Information Bottleneck

A hallmark of chaotic dynamics is the loss of information with time. Although information loss is often expressed through a connection to Lyapunov exponents -- valid in the limit of high information about the system state -- this picture misses the rich spectrum of information decay across different levels of granularity. Here we show how machine learning presents new opportunities for the study of information loss in chaotic dynamics, with a double pendulum serving as a model system. We use the Information Bottleneck as a training objective for a neural network to extract information from the state of the system that is optimally predictive of the future state after a prescribed time horizon. We then decompose the optimally predictive information by distributing a bottleneck to each state variable, recovering the relative importance of the variables in determining future evolution. The framework we develop is broadly applicable to chaotic systems and pragmatic to apply, leveraging data and machine learning to monitor the limits of predictability and map out the loss of information.

混沌动力学的一个标志是信息随时间的丢失。尽管信息丢失通常通过与李雅普诺夫指数的联系来表述——这在对系统状态有高度信息的极限情况下是有效的——但这种描述忽略了不同粒度级别上信息衰减的丰富频谱。在这里，我们展示了机器学习如何为研究混沌动力学中的信息丢失提供新的机会，以双摆作为一个模型系统。我们将信息瓶颈作为神经网络的训练目标，从系统状态中提取信息，这些信息在规定的时间范围内对未来状态具有最优的预测性。然后，我们通过给每个状态变量分配一个瓶颈来分解最优预测信息，恢复变量在决定未来演化中的相对重要性。我们开发的框架广泛适用于混沌系统，且应用起来很实用，它利用数据和机器学习来监测可预测性的极限并绘制出信息丢失的情况。