Why Is the Three-Body Problem Unsolvable?
Researchers have solved a set of simple examples of the chaotic three-body problem.
Space travel and most real-life systems are chaotic, making this research valuable.
Neural networks have the potential to solve, or at least model, chaotic problems better than traditional supercomputers.
Computer scientists have shown that a special neural network is likely able to solve simpler exemplars of the chaotic three-body problem, reports Tech Xplore. If the results seem hard to parse, that’s because the three-body problem and its implications are also pretty hard to parse.
The three-body problem is an arm of cosmology, where the “bodies” are celestial, like calculating where planets are in relation to each other over time. (Chinese sci-fi author Liu Cixin used the term as a pun for the title of his Hugo Award-winning 2015 novel about murdered astrophysicists.) Applications range from the earliest low-tech ship navigators to modern theories of spaceflight like gravity assists, and the mathematical complexity of the problem itself has made it interesting to both mathematicians and computer scientists for many years.
We sometimes think of space as empty because of popular misconceptions, but even in the relative vacuum, space is filled with clashing gravity fields, magnetic fields, solar winds (a misnomer, because there’s no air), and more. Everything is pushed and pulled by different forces—so many forces and with such complexity that the “three bodies” are almost completely unpredictable from moment to moment, even if we know where they just were an instant before.
In the past, scientists relegated “chaotic” behaviors like the three-body problem to a dusty island for misfit problems. As supercomputers grew more powerful, these scientists realized they could use rapidly increasing computing power to sledgehammer at complicated math problems. In turn, artificial neural networks offer a step up from simply supercomputing.
These machines, inspired by real biological processes found in nature, can more closely model chaos because of their capacity to work on nonlinear problems. When we (or even monkeys and other nonhuman primates) see someone place two items together, we don’t expect to find three items as the sum. This is a linear process of sorts, where we use our knowledge of the inputs to predict proportional outputs.
In a nonlinear system like the chaotic three-body problem, all bets are off, and our intuitions are scrambled. Much of applied science involves nonlinear systems and problem solving. So far, scientists haven’t succeeded in solving the three-body problem except in very defanged formats: the two-body problem is solved, and scientists can solve what they call a “restricted” three-body problem, which is when one body is so negligible in mass that it basically disappears into the equation.
Think of this like taking the derivative of an equation, when a constant simply falls away to become 0—or calculating n-complexity in computer science, when the exponent or log is usually all that matters, and other information is discarded.
All of this means that a neural network that can model and solve even simple forms of the chaotic three-body problem, where all three bodies are statistically significant independent actors, is a huge deal. These researchers—from the University of Edinburgh, the University of Cambridge, Campus Universitario de Santiago, and Leiden University—pitted their neural network against a traditional supercomputer trained to solve simpler three-body problems, and they say their network solved these examples much, much faster.
There are complications, though. Having an existing, specialized supercomputer give them the answers to their examples means the scientists had a ready answer key to check against—without this resource, it’s not clear a neural network would readily generate correct answers on its own, especially as problems got more complex. Deep machine learning like that used by neural networks is something of a black box.
Moreover, the researchers themselves conclude that their neural network approximated the concrete results from the traditional supercomputer. Like using 3.14 instead of pi itself, this kind of application almost always has caveats. The new research is exciting, but it doesn’t represent a clear and concrete step forward without a lot more context and outside input.
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