Using artificial intelligence, an international team of physicists has shown that the thousands of equations needed to model a complex system of interacting electrons can be reduced to just four. This was done by using machine learning to identify patterns previously hidden within the system of equations. The technique could be used to vastly reduce the effort required to calculate electronic properties, says the team, which was led by Domenico Di Sante at the University of Bologna, who is also a visiting research fellow at the Flatiron Institute in New York City.

Quantum interactions between electrons underly the properties of matter, and over the past century physicists have developed mathematical and computational tools to boost our understanding of systems ranging from individual atoms to solid materials. These models must consider entanglement, a quantum phenomenon that allows stronger correlations between electrons than exists in classical physics.

A powerful mathematical tool for studying how quantum interactions between electrons in a material affect the macroscopic properties of the material is the renormalization group. However, this approach still comes with enormous challenges associated with solving large systems of coupled differential equations. Indeed, thousands, or even millions of equations may be required.

### Hop to it

In their study, Di Sante’s team considered how the complexity of the renormalization group could be reduced by using machine learning to spot patterns hidden within large groups of equations – patterns that have escaped the notice of human researchers. To explore this idea, they considered the idealized 2D Hubbard model in which electrons “hop” between adjacent lattice sites in a solid material.

In this model, transitions between conducting and insulating electron systems are simulated by adjusting parameters that describe two competing processes: one that encourages the quantum tunnelling (hopping) of electrons between neighbouring lattice sites; and the other reflecting the fact that multiple electrons do not want to occupy the same lattice site.

As electrons interact with each other, they become entangled. This entanglement persists over long distances and must be accounted for in the coupled differential equations describing the system – making the equations very difficult to solve using renormalization group techniques.

### Identifying redundancies

To solve the renormalization group of this model, Di Sante and colleagues first trained an artificial neural network to recognize underlying patterns within hundreds of thousands of differential equations. By identifying redundancies within multiple equations, their algorithm sought to reduce the problem to a far smaller group of equations. After weeks of training, the algorithm reduced the problem down to just four equations, and the team says this was done without sacrificing any accuracy in their solutions.

Artificial intelligence simplifies calculations of electronic properties

The researchers hope that their hugely successful result could soon be readily applied to quantum problems beyond the Hubbard model. This could allow researchers to model quantum states of matter such as superconductivity with far greater computational efficiency. This could in turn lead to designs of exotic new materials. By investigating the patterns picked up by the artificial neural network, Di Sante’s team also hope that physicists may gain deeper insights into quantum effects that have evaded physicists so far.

The research is described in *Physical Review Letters*.