Solving problems of analytic continuation using deep learning

A general framework for building an artificial neural network (ANN) that solves this task with a supervised learning approach.

Inverse problems are addressed in numerous areas of physics, with the analytic continuation of the imaginary Green’s function into the real frequency domain being an especially significant example. Be that as it may, the analytic continuation problem is poorly characterized, and presently, no analytic change for unraveling it is known.

As a part of the semester project, an EPFL student named Romain Fournier applied machine learning to the problem of analytic continuation. He has shown that deep learning can be used to analytically connect digital simulations and experimental results more quickly and reliably than conventional methods.

Fournier said“The main challenge in the translation process is that there’s an unlimited number of mathematical solutions to a given problem. It’s a little like, instead of being asked what 2+2 equals, you were asked what math operation answers 4. Among the many possible answers, we’re only interested in the one that makes sense in the physical world. So we’re talking about an ill-defined problem, which is a common situation in science.”

This new approach involves teaching a neural network to run the translation process by feeding it examples of data simulations that could be obtained experimentally.

Fournier said, “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.”

Unlike conventional methods, this new method provides more reliable answers than traditional methods and does so more quickly.

Solving problems of analytic continuation using deep learning

Oleg Yazyev, an assistant professor, said, “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. 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.”

The study is published in the journal Physical Review Letters.

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