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Uncertain Math Might Be Better Math

To make models of natural systems more accurate and efficient, make them more random.
Wake turbulence. Image: Bernal Saborio/Flickr

Precision almost seems like part of the definition of math. The realm of numbers and symbols and methods seems to be one in which vagueness is an outcast.

But maybe, in all of this precision-seeking, we're going astray. This is the suggestion behind new research from mathematicians at Brown University: for more efficient algorithms and modeling techniques, we need to add some uncertainty.

The problem the Brown researchers approached was modeling certain natural systems, particularly ones involving fluid dynamics. Turbulence or shock waves within these systems are often described using Burgers' equations, which look like this:

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Understanding differential equations isn't a prerequisite for reading Motherboard, but you can still get the general idea. Given values for viscosity ( v)—how thick or sludgy a fluid is—and how fast it's moving, e.g. velocity (u), you can work out interesting and helpful things about how how that fluid responds to some sort of disturbance.

"Say you have a wave that's moving very fast in the atmosphere," explains George Karniadakis, the Brown University mathematics professor behind the research, in a statement. "If the rest of the air in the domain is at rest, then flow one goes over the other. That creates a very stiff front or a shock, and that's what Burgers' equation describes."

I posted the actual equations above less so that you'd take them in and go "Ahhhhh, I see" and more to illustrate Karniadakis' main point that these equations are rather sterilized; they describe something very messy in conspicuously unmessy terms. There is more to air flow in the atmosphere than that.

A model can only describe something based on what it knows, however, and with fluids in particular there are usually many things that go unknown. With Karniadakis' airflow example above, there is also the effect of the ground terrain below—mountains, plains, canyons, all the rest of it.

Just ignoring these unknowns seems natural enough in math. The other alternative would be to make something up, and how scientific is that? Karniadakis' suggestion fits into a third category, one that's already familiar enough to computer graphics people used to modeling textures and natural systems: randomness.

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Rather than imagining the ground below has exactly zero influence, we can imagine that it has an unknown influence described by random numbers, or at least random numbers within some reasonable parameters.

Noise is more precise than nothing

This is an emerging idea in mathematics known as uncertainty quantification (UQ). "The general idea in UQ," Karniadakis explains, "is that when we model a system, we have to simplify it. When we simplify it, we throw out important degrees of freedom. So in UQ, we account for the fact that we committed a crime with our simplification and we try to reintroduce some of those degrees of freedom as a random forcing. It allows us to get more realism from our simulations and our predictions."

Forcing in differential equations is a way of introducing a kind of interloper equation or term. While the "main" equation might still exist as the Burgers' equation we know and love, forcing can be applied as an outside influence. It's part of the equation, but only as kind of an annex of it.

So, this forcing term would be some randomness. Not complete chaos, but a sort of randomness tailored to the environment or based on an observed average. Like, we won't know exactly what terrain we have under our air masses, but we can make an average for a region, and from that average we can make a suitable random distribution. That's a satisfying notion, eh? Noise is more precise than nothing.