You don’t need to know everything to make good predictions

by Rayan Moussa

Imagine yourself attempting to predict the future of a large crowd of people, but all you have is a few friends scattered around watching the sea of people in only a select few locations. Some of the people there are shouting, some are quiet, some dart around while others barely move. It is practically impossible to observe everyone, but you still want to guess where things are likely to go. What would be the best way to analyze such a crowd given the limited information your friends can give you? What should they look at? Where should they be located? Although the particular systems that physicists, mathematicians and other researchers typically study might differ from the crowd you’re trying to describe, the underlying problem is exactly the same. Having only limited information about a system with a large number of dynamical elements, how do we approach the problem of describing it as accurately as possible? Furthermore, one may even wonder, how could this even be possible? Shouldn’t the simple fact that we don’t have access to much of the current information that the system possesses restrict us from predicting with accuracy what the system’s current and future behavior looks like?

Fortunately, physicists were able to tackle these issues by developing a mathematical theory, now famously known as the Mori-Zwanzig formalism, which explains the way to systematically treat very large and complex systems in terms of fewer and simpler elements, allowing us to not only construct new methods of solving such problems, but also understand how similar classical problems, such as those of statistical mechanics and high-energy physics, were able to be resolved in the first place. In recent years, this idea has been popping up in many other places as well, such as numerical simulations, artificial intelligence/machine learning and even in discussions about the philosophy of science itself.

The surprising way this works, as Hajime Mori and Robert Zwanzig showed back in the 1960s, requires a deeper apprehension of how information itself operates, and their formalism was, and still is, of crucial importance to the understanding of how one should think of such coarse-grained systems in general. The key insight, which allows their formalism to function effectively, is both simple and elegant. If you have limited access to spatial information about your system, as would be the case if you observe it from only a few spatial locations, you can compensate for this lack of information by using more temporal information instead, which means that by recording information about the few spatial locations you have access to over a sufficiently lengthy time, you should be able to predict the behavior of the system with increasingly higher and higher accuracy. Their formalism not only proves that this is possible, but also offers the instructions of what information to record, where to record it and how to translate it into a form which tells you more about the current state of the system and, of course, it’s future behavior.

The promising application of the Mori-Zwanzig theory lies in improving the efficiency of complex computer simulations, which is the primary focus of my research. This theory allows us to streamline extensive, time consuming calculations by concentrating on the most significant system components, akin to zooming out to get an overview without dwelling on the unnecessary details. A strongly related concept has existed since the 1970s in the form of Multigrid Methods, which accelerate the resolution of specific mathematical problems by operating at varying levels of detail. Researchers are only now beginning to merge the strengths of the Mori-Zwanzig theory with these traditional methods, with the aim of developing more intelligent, faster algorithms that outperform previous solutions, particularly for complex problems that are typically slow or costly to tackle.

This emerging direction holds the potential to greatly improve the area of scientific computing, allowing us to more effectively model phenomena ranging from weather systems to material design. It’s only a matter of time before additional fields, such as theoretical physics, numerical linear algebra, artificial intelligence, and machine learning, capitalize on the profound insights offered by this fascinating and powerful theory.

Video of the opening of the 2023 edition of the San Fermin festival, in PamPamplona, Spain. © Bartolo Lab, directed by Denis Bartolo, Laboratoire de Physique (LPENSL, CNRS / ENS de Lyon), Lyon, France.