Electrons whizzing through a grid-like lattice don't behave at all like pretty silver spheres in a pinball machine. They blur and bend in collective dances, following whims of a wave-like reality that are hard enough to imagine, let alone compute.

And yet scientists have succeeded in doing just that, capturing the motion of electrons moving about a square lattice in simulations that – until now – had required hundreds of thousands of individual equations to produce.

Using artificial intelligence (AI) to reduce that task down to just four equations, physicists have made their job of studying the emergent properties of complex quantum materials a whole lot more manageable.

In doing so, this computing feat could help tackle one of the most intractable problems of quantum physics, the 'many-electron' problem, which attempts to describe systems containing large numbers of interacting electrons.

It could also advance a truly legendary tool for predicting electron behavior in solid state materials, the Hubbard model – all the while bettering our understanding of how handy phases of matter, such as superconductivity, occur.

Superconductivity is a strange phenomenon that arises when a current of electrons flow unimpeded through a material, losing next to no energy as they slip from one point to another. Unfortunately most practical means of creating such a state rely on insanely low temperatures, if not ridiculously high pressures. Harnessing superconductivity closer to room temperature could lead to far more efficient electricity grids and devices.

Since achieving superconductivity under more reasonable conditions remains a lofty goal, physicists have taken to using models to predict how electrons could behave under various circumstances, and therefore which materials make suitable conductors or insulators.

These models have their work cut out for them. Electrons don't roll through the network of atoms like tiny balls, after all, with clearly defined positions and trajectories. Their activity is a mess of probability, influenced not only by their surroundings but by their history of interactions with other electrons they've bumped into on the way.

When electrons interact, their fates can become intimately intertwined, or 'entangled'. Simulating the behavior of one electron means tracking the range of possibilities of all electrons in a model system at once, which makes the computational challenge exponentially harder.

The Hubbard model is a decades-old mathematical model that describes the confusing motion of electrons through a lattice of atoms somewhat accurately. Over the years and much to physicists' delight, the deceptively simple model has been experimentally realized in the behavior of a wide array of complex materials.

With ever-increasing computer power, researchers have developed numerical simulations based on Hubbard model physics that allow them to get a grip on the role of the topology of the underlying lattice.

In 2019, for instance, researchers proved the Hubble Model was capable of representing superconductivity higher-than-ultra-cold temperatures, giving the green light to researchers to use the model for deeper insights into the field.

This new study could be another big leap, greatly simplifying the number of equations required. Researchers developed a machine-learning algorithm to refine a mathematical apparatus called a renormalization group, which physicists use to explore changes in a material system when properties such as temperature are altered.

"It's essentially a machine that has the power to discover hidden patterns," physicist and lead author Domenico Di Sante, of the University of Bologna in Italy, says of the program the team developed.

"We start with this huge object of all these coupled-together differential equations" – each representing pairs of entangled electrons – "then we're using machine learning to turn it into something so small you can count it on your fingers," Di Sante says of their approach.

The researchers demonstrated that their data-driven algorithm could efficiently learn and recapitulate dynamics of the Hubbard model, using only a handful of equations – four to be precise – and without sacrificing accuracy.

"When we saw the result, we said, 'Wow, this is more than what we expected.' We were really able to capture the relevant physics," says Di Sante.

Training the machine learning program using data took weeks, but Di Sante and colleagues say it could now be adapted to work on other, tantalizing condensed-matter problems.

The simulations thus far only capture a relatively small number of variables in the lattice network, but the researchers expect their method should be fairly scalable to other systems.

If so, it could in the future be used to probe the suitability of conducting materials for applications that include clean energy generation, or to aid in the design of materials that may one day deliver that elusive room-temperature superconductivity.

The real test, the researchers note, will be how well the approach works on more complex quantum systems such as materials in which electrons interact at long distances.

For now, the work demonstrates the possibility of using AI to extract compact representations of dynamic electrons, "a goal of utmost importance for the success of cutting-edge quantum field theoretical methods for tackling the many-electron problem," the researchers conclude in their abstract.

The research was published in *Physical Review Letters.*