标题： 来自不可预测混沌的机器学习预测 显示英文标题

标题： Machine learning predictions from unpredictable chaos

摘要： 混沌在自然界中无处不在，对其理解带来了巨大的社会和经济效益。然而，由于混沌系统对初始条件的敏感性、非周期性行为、分数维、非线性和奇怪吸引子等特性，混沌系统的不可预测性已成为教科书上的概念。 在这项工作中，我们首次引入了混沌学习，这是一种新颖的多尺度拓扑范式，能够从混沌系统中进行准确的预测。我们表明，看似随机且不可预测的混沌动力学出人意料地提供了前所未有的定量预测。 具体而言，我们设计了多尺度拓扑拉普拉斯算子，将现实世界的数据嵌入到一系列交互式的混沌动力系统中，调节它们的动力学行为，并实现对输入数据的准确预测。作为一个概念验证，我们考虑了来自四个类别实际问题的28个数据集：10个脑电波、四个基准蛋白质数据集、13个单细胞RNA测序数据集和一个图像数据集，以及两个不同的混沌动力系统，即洛伦兹和罗素吸引子。 我们展示了从混沌中预测物理属性的混沌学习预测。我们的新混沌学习范式深刻改变了教科书中关于混沌的传统认知，并首次将拓扑、混沌和学习联系起来。 显示英文摘要

摘要： Chaos is omnipresent in nature, and its understanding provides enormous social and economic benefits. However, the unpredictability of chaotic systems is a textbook concept due to their sensitivity to initial conditions, aperiodic behavior, fractal dimensions, nonlinearity, and strange attractors. In this work, we introduce, for the first time, chaotic learning, a novel multiscale topological paradigm that enables accurate predictions from chaotic systems. We show that seemingly random and unpredictable chaotic dynamics counterintuitively offer unprecedented quantitative predictions. Specifically, we devise multiscale topological Laplacians to embed real-world data into a family of interactive chaotic dynamical systems, modulate their dynamical behaviors, and enable the accurate prediction of the input data. As a proof of concept, we consider 28 datasets from four categories of realistic problems: 10 brain waves, four benchmark protein datasets, 13 single-cell RNA sequencing datasets, and an image dataset, as well as two distinct chaotic dynamical systems, namely the Lorenz and Rossler attractors. We demonstrate chaotic learning predictions of the physical properties from chaos. Our new chaotic learning paradigm profoundly changes the textbook perception of chaos and bridges topology, chaos, and learning for the first time.