Coloring a chaotic attractor with machine learning: Optimized measurements of chaotic dynamical systems via the information bottleneck

Nonlinearity can fundamentally limit predictability in a dynamical system, with deterministic chaos serving as a minimal reproducible example. Predictability is curtailed by a finite rate of information destruction—and commensurate information creation—as the dynamical system evolves. Is it possible for a measurement of finite capacity to capture all of the ephemeral information? Remarkably, such measurements do exist and take the form of highly specific partitions, or colorings, of the chaotic attractor. These optimal measurements are a lossy compression of each continuous-valued state of the dynamical system that yield a lossless compression of a trajectory when measured repeatedly ad infinitum. Here we establish an equivalence between the definition of an optimal measurement and an objective from rate-distortion theory, the distributed information bottleneck. We leverage machine learning to optimize measurement schemes given trajectory data and obtain partitions for several chaotic systems that capture nearly all of the information created by the dynamics. Traditionally having been found by leveraging intimate knowledge of the system, the measurements illuminate the fundamental barrier to prediction imposed by the dynamics.