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A mathematician at Lawrence Livermore National Laboratory (LLNL), along with collaborators at the University of Massachusetts, Dartmouth and the University of Mississippi, has developed a machine-learning technique to determine the motion of binary black holes using gravitational wave data.

Scientists believe binary black holes orbit each other for billions of years before eventually colliding to form a single massive black hole. During the final moments, their mass is converted to a gigantic burst of energy that can be detected in the form of gravitational waves. The Laser Interferometer Gravitational-Wave Observatory (LIGO) first detected these gravitational waves in 2016, opening the door to a better understanding of the motion of black holes and other spacetime-warping phenomena.

Until now, studying that motion has required applying simplified field equations first proposed by Albert Einstein and then solving them to calculate the emitted gravitational waves; a complex and time-consuming approach that requires the use of supercomputers or approximations that can lead to errors, or even break down completely when applied to more complicated black hole systems.

However, Brendan Keith, a postdoctoral researcher in LLNL’s Center for Applied Scientific Computing, and his collaborators took an inverse approach to the problem. Working backwards, they used gravitational wave data and machine-learning techniques to create a mathematical model that can generate differential equations describing the dynamics of merging black holes for a range of cases. Using a laptop computer, their waveform inversion strategy can quickly produce equations for the motion of binary black holes that are as accurate as equations that previously took years to develop or models that can take weeks to run on supercomputers.

“We have all this data that relates to more complicated black hole systems, and we don’t have complete models to describe the full range of these systems, even after decades of work,” Keith said. “Machine learning will tell us what the equations are automatically. It will take in your data and it will output an equation in a few minutes to an hour, and that equation might be as accurate as something a person had been working on for 10 or 20 years.”

Keith was the lead author for the study, which appeared in the journal Physical Review Research. The approach only requires the application of Kepler’s laws of planetary motion, instead of Einstein’s more complex general theory of relativity theory, and the math needed to solve an inverse problem.

“We had some confidence that if we went from one dimension to one dimension, it would work — that’s what the earlier papers had done — but a gravitational wave is lower dimensional data than the trajectory of a black hole,” Keith said. “It was a big, exciting moment when we found out it does work.”

The work was performed with a grant from the National Science Foundation and funding from LLNL.