Ruehle and collaborators took up the old problem of approximating Calabi-Yau metrics. Anderson and others also revitalized their earlier attempts to overcome step 2. The physicists found that neural networks provided the speed and flexibility that earlier techniques had lacked. The algorithms were able to guess a metric, check the curvature at many thousands of points in 6D space, and repeatedly adjust the guess until the curvature vanished all over the manifold. All the researchers had to do was tweak freely available machine learning packages; by 2020, multiple groups had released custom packages for computing Calabi-Yau metrics.

With the ability to obtain metrics, physicists could finally contemplate the finer features of the large-scale universes corresponding to each manifold. “The first thing I did after I had it, I calculated masses of particles,” Ruehle said.

From Strings to Quarks

In 2021, Ruehle, collaborating with Ashmore, cranked out the masses of exotic heavy particles that depend only on the curves of the Calabi-Yau. But these hypothetical particles would be far too massive to detect. To calculate the masses of familiar particles like electrons — a goal string theorists have chased for decades — the machine learners would have to do more.

Lightweight matter particles acquire their mass through interactions with the Higgs field, a field of energy that extends throughout space. The more a given particle takes notice of the Higgs field, the heavier it is. How strongly each particle interacts with the Higgs is labeled by a quantity called its Yukawa coupling. And in string theory, Yukawa couplings depend on two things. One is the metric of the Calabi-Yau manifold, which is like the shape of the doughnut. The other is the way quantum fields (arising as collections of strings) spread out over the manifold. These quantum fields are a bit like sprinkles; their arrangement is related to the doughnut’s shape but also somewhat independent.

Ruehle and other physicists had released software packages that could get the doughnut shape. The last step was to get the sprinkles — and neural networks proved capable of that task, too. Two teams put all the pieces together earlier this year.

An international collaboration led by Challenger Mishra of the University of Cambridge first built on top of Ruehle’s package to calculate the metric — the geometry of the doughnut itself. Then they used homegrown neural networks to compute the way the quantum fields overlap as they curve around the manifold, like the doughnut’s sprinkles. Importantly, they worked in a context where the geometry of the fields and that of the manifold are tightly linked, a setup in which the Yukawa couplings are already known. When the group calculated the couplings with the neural networks, the results matched the known answers.

“People have been wanting to do this since before I was born in the ’80s,” Mishra said.

A group led by string theory veterans Burt Ovrut of the University of Pennsylvania and Andre Lukas of Oxford went further. They too started with Ruehle’s metric-calculating software, which Lukas had helped develop. Building on that foundation, they added an array of 11 neural networks to handle the different types of sprinkles. These networks allowed them to calculate an assortment of fields that could take on a richer variety of shapes, creating a more realistic setting that can’t be studied with any other techniques. This army of machines learned the metric and the arrangement of the fields, calculated the Yukawa couplings, and spit out the masses of three types of quarks. It did all this for six differently shaped Calabi-Yau manifolds. “This is the first time anybody has been able to calculate them to that degree of accuracy,” Anderson said.

None of those Calabi-Yaus underlies our universe, because two of the quarks have identical masses, while the six varieties in our world come in three tiers of masses. Rather, the results represent a proof of principle that machine learning algorithms can take physicists from a Calabi-Yau manifold all the way to specific particle masses.

“Until now, any such calculations would have been unthinkable,” said Constantin, a member of the group based at Oxford.

Numbers Game

The neural networks choke on doughnuts with more than a handful of holes, and researchers would eventually like to study manifolds with hundreds. And so far, the researchers have considered only rather simple quantum fields. To go all the way to the Standard Model, Ashmore said, “you might need a more sophisticated neural network.”

Bigger challenges loom on the horizon. Attempting to find our particle physics in the solutions of string theory — if it’s in there at all — is a numbers game. The more sprinkle-laden doughnuts you can check, the more likely you are to find a match. After decades of effort, string theorists can finally check doughnuts and compare them with reality: the masses and couplings of the elementary particles we observe. But even the most optimistic theorists recognize that the odds of finding a match by blind luck are cosmically low. The number of Calabi-Yau doughnuts alone may be infinite. “You need to learn how to game the system,” Ruehle said.

One approach is to check thousands of Calabi-Yau manifolds and try to suss out any patterns that could steer the search. By stretching and squeezing the manifolds in different ways, for instance, physicists might develop an intuitive sense of what shapes lead to what particles. “What you really hope is that you have some strong reasoning after looking at particular models,” Ashmore said, “and you stumble into the right model for our world.”

Lukas and colleagues at Oxford plan to start that exploration, prodding their most promising doughnuts and fiddling more with the sprinkles as they try to find a manifold that produces a realistic population of quarks. Constantin believes that they will find a manifold reproducing the masses of the rest of the known particles in a matter of years.

Other string theorists, however, think it’s premature to start scrutinizing individual manifolds. Thomas Van Riet of KU Leuven is a string theorist pursuing the “swampland” research program, which seeks to identify features shared by all mathematically consistent string theory solutions — such as the extreme weakness of gravity relative to the other forces. He and his colleagues aspire to rule out broad swaths of string solutions — that is, possible universes — before they even start to think about specific doughnuts and sprinkles.

“It’s good that people do this machine learning business, because I’m sure we will need it at some point,” Van Riet said. But first “we need to think about the underlying principles, the patterns. What they’re asking about is the details.”

Plenty of physicists have moved on from string theory to pursue other theories of quantum gravity. And the recent machine learning developments are unlikely to bring them back. Renate Loll, a physicist at Radboud University in the Netherlands, said that to truly impress, string theorists will need to predict — and confirm — new physical phenomena beyond the Standard Model. “It is a needle-in-a-haystack search, and I am not sure what we would learn from it even if there was convincing, quantitative evidence that it is possible” to reproduce the Standard Model, she said. “To make it interesting, there should be some new physical predictions.”

New predictions are indeed the ultimate goal of many of the machine learners. They hope that string theory will prove rather rigid, in the sense that doughnuts matching our universe will have commonalities. These doughnuts might, for instance, all contain a kind of novel particle that could serve as a target for experiments. For now, though, that’s purely aspirational, and it might not pan out.

“String theory is spectacular. Many string theorists are wonderful. But the track record for qualitatively correct statements about the universe is really garbage,” said Nima Arkani-Hamed, a theoretical physicist at the Institute for Advanced Study in Princeton, New Jersey.

Ultimately, the question of what string theory predicts remains open. Now that string theorists are leveraging the power of neural networks to connect the 6D microworlds of strings with the 4D macroworlds of particles, they stand a better chance of someday answering it.

“Without a doubt, there are loads of string theories that have nothing to do with nature,” Anderson said. “The question is: Are there any that do have something to do with it? The answer might be no, but I think it’s really interesting to try to push the theory to decide.”