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Igor Mezić applies Koopman Operator theory to understanding dynamical systems — like the 405 freeway
raffic has been vexing drivers since not long after the first Model T’s rolled off Henry Ford’s assembly line. And while the cause of traffic may seem obvious — too many cars using a stretch of roadway at a certain time — the dynamics that lead to congestion are not. That’s because traffic is, as described by Igor Mezić, professor of mechanical engineering at UC Santa Barbara, a dynamical system, that is, one that is in constant flux, with an almost infinite number of ever-changing inputs dictating how efficiently the system functions at any given time. In a paper to be published in an upcoming issue of the journal Nature Communications, Mezić and Allan Avila, a fifth-year PhD student in his lab, describe their research, which makes use of Los Angeles freeways, some of the busiest in the nation, as a test case for a new approach to understanding and forecasting traffic patterns. Their method employs a sophisticated mathematical approach known as Koopman Operator theory, a subject on which Mezić is a widely recognized expert. The research was funded as part of a $6.5 million U.S. Army Multidisciplinary University Research Initiative (MURI) project focused on developing methods to describe and predict the behavior of dynamical systems generally. Traffic is such a system, as noted above, and the data was available, so, Mezić and Avila took on the challenge. For decades, says Avila, the approach to trying to understand traffic as a dynamical system has been “to write some sort of evolution equations to describe how traffic dynamics evolve in time.” Avila explains the problem with that approach: “Whenever
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you’re trying to develop such a mathematical model of a system, you make assumptions that allow you to simplify your equations. For example, you may assume that the highway system consists of only a single lane and has no exits or entrances.” In other words, you get an incomplete picture of the reality of the dynamical system. There are good reasons for that, Avila says. “It’s very difficult to get those equations right even for a single-lane road, and it’s much more difficult to get a multi-lane description correct, because then you have to account for the complexities of lane-changing behavior and vehicles exiting and entering.” The Koopman Operator method requires no assumptions and enables accurate short-term predictions based on limited realtime data. That is in contrast to machine-learning (ML) techniques, considered today’s state-of-the-art for traffic modeling, which require large amounts of training data to learn traffic dynamics. Unfortunately, even with voluminous data, they are contextually limited. Mezić notes, “If you show one of these machine-learning methods a stretch of freeway in L.A. and then try to use the same model to apply what you learned somewhere in San Francisco or Denver, often times it starts to perform poorly, and you need to retrain it based on data for the new location.” Mezić and Avila’s method overcomes several problems inherent in mathematical and machine-learning models, the most obvious one being that forecasts often fail to match what actually happens. “With our method, we don’t need to write down equations explicitly,” Avila explains. “Instead, we take observations of the dynamical system