Solving problems of analytic continuation through machine learning

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Solving problems of analytic continuation through machine learning

An EPFL student has shown how deep learning can be used to analytically connect digital simulations and experimental results more quickly and reliably than conventional methods. This work, which the student carried out for his semester project, was recently published in Physical Review Letters.

It’s not unusual for scientists to compare experimental results with the predictions made by theoretical models. And nowadays, generating these predictions usually involves digital methods. For physicists, these methods sometimes require them to work in what they call imaginary time, where their results must be translated before they can be compared to laboratory data. This translation process is referred to as analytic continuation.

As part of his semester project, which he did in the lab of EPFL’s Chair of Computational Condensed Matter Physics (C3MP), Romain Fournier applied machine learning to the problem of analytic continuation. His findings were just published in the leading physics journal Physical Review Letters - which is highly unusual for an undergraduate project.

Prior knowledge

"The main challenge in the translation process is that there’s an unlimited number of mathematical solutions to a given problem," says Fournier, who recently began his doctoral studies in statistics at the University of Oxford. "It’s a little like, instead of being asked what 2+2 equals, you were asked what math operation gives an answer of 4. Among the many possible answers, we’re obviously only interested in the one that makes sense in the physical world. So we’re talking about an ill-defined problem, which is a very common situation in science." And the only way to solve this problem is to bring in prior knowledge.

Fournier’s approach consisted of teaching a neural network to run the translation process by feeding it examples of data simulations that could be obtained experimentally. "It’s very easy to go from experimental data to imaginary-time data, and so we were able to quickly build up a big database that we could use to train our model," says Fournier. This approach was suggested to him by Quansheng Wu, a postdoc at the C3MP lab who supervised Fournier with the support of NCCR MARVEL, a research center created to design and discover new materials through supercomputer-powered simulations.

And so while conventional methods factor in prior knowledge indirectly, this new method does it in a more straightforward way by generating results that are similar to those that would be achieved experimentally. The neural network, once it has been fed these results, provides more reliable answers than conventional methods and does so more quickly.

Growing influence

For Oleg Yazyev, an assistant professor and head of the C3MP, machine-learning tools will have a growing influence on the field of physics in the coming years. "The fact that an ordinary semester project can turn into a scientific advancement worthy of being published in a leading journal is a real source of motivation for our students," says Yazyev. "That doesn’t happen every day, of course. But when it does, it provides our up-and-coming researchers with a real career boost. On top of that, more experienced researchers like me can use these kinds of projects to test some of our crazier ideas."