Half a century ago, the pioneers of chaos theory discovered that the “butterfly effect” makes long-term prediction impossible. Even the smallest perturbation to a complex system (like the weather, the economy or just about anything else) can touch off a concatenation of events that leads to a dramatically divergent future.
In a series of results reported in the journals Physical Review Letters and Chaos, scientists have used machine learning — the same computational technique behind recent successes in artificial intelligence — to predict the future evolution of chaotic systems out to stunningly distant horizons. The approach is being lauded by outside experts as groundbreaking and likely to find wide application.
The findings come from veteran chaos theorist Edward Ott and four collaborators at the University of Maryland. They employed a machine-learning algorithm called reservoir computing to “learn” the dynamics of an archetypal chaotic system called the Kuramoto-Sivashinsky equation. The evolving solution to this equation behaves like a flame front, flickering as it advances through a combustible medium. The equation also describes drift waves in plasmas and other phenomena, and serves as “a test bed for studying turbulence and spatiotemporal chaos,” said Jaideep Pathak, Ott’s graduate student and the lead author of the new papers.
The Lyapunov time represents how long it takes for two almost-identical states of a chaotic system to exponentially diverge. As such, it typically sets the horizon of predictability. The duration of a Lyapunov time varies for different systems, from milliseconds to millions of years. (It’s a few days in the case of the weather.) The shorter it is, the touchier or more prone to the butterfly effect a system is, with similar states departing more rapidly for disparate futures.
After training itself on data from the past evolution of the Kuramoto-Sivashinsky equation, the researchers’ computer could then closely predict how the flamelike system would continue to evolve out to eight “Lyapunov times” into the future, eight times further ahead than previous methods allowed, loosely speaking.
The algorithm knows nothing about the Kuramoto-Sivashinsky equation itself; it only sees data recorded about the evolving solution to the equation. This makes the machine-learning approach powerful; in many cases, the equations describing a chaotic system aren’t known, crippling dynamicists’ efforts to model and predict them. Ott and company’s results suggest you don’t need the equations — only data. “This paper suggests that one day we might be able perhaps to predict weather by machine-learning algorithms and not by sophisticated models of the atmosphere,” Kantz said.
Exactly why the algorithm is so good at learning the dynamics of chaotic systems is not yet well understood, beyond the idea that the computer tunes its own formulas in response to data until the formulas replicate the system’s dynamics. The technique works so well, in fact, that Ott and some of the other Maryland researchers now intend to use chaos theory as a way to better understand the internal machinations of neural networks.
Besides weather forecasting, experts say the machine-learning technique could help with monitoring cardiac arrhythmias for signs of impending heart attacks and monitoring neuronal firing patterns in the brain for signs of neuron spikes. More speculatively, it might also help with predicting rogue waves, which endanger ships, and possibly even earthquakes.