The renormalization group is a fundamental tool in physics that allows us to understand how systems change depending on the length scale at which we observe them. Traditionally, this process is one-way: we “blur” the image of a system, gradually removing microscopic details to reveal a broader, macroscopic picture.
In a recently published study in Physical Review E, our researchers Tomislav Ivek and Ivan Balog, as part of an international collaboration, posed the inverse question: can a neural network do the opposite—reconstruct fine, microscopic details starting from coarse-grained information?
As a test case, they used the two-dimensional Ising model (a standard model in physics that describes magnetism at the microscopic level) at the critical point, where a system displays complex patterns across all scales simultaneously. The researchers trained very simple neural networks to learn how to generate fine spin configurations from “coarse” data.
A surprising result is that even a minimal model – a single-layer network with only three tunable parameters, can, through an iterative local rule, “dream up” realistic critical configurations starting from a single initial spin. In other words, the network did not rely on Monte Carlo simulations; instead, it independently discovered the simple rules that govern the structure of the critical system.
The generated samples successfully reproduce a range of physical properties of critical systems, such as the scaling of magnetic susceptibility and heat capacity. Interestingly, increasing the complexity of the neural network did not lead to better results, suggesting that the fundamental structure of these complex physical systems can be captured by simple local rules.
This work highlights the connection between physical universality in critical phenomena (where different systems exhibit identical behavior near phase transitions) and the ability of minimal machine learning models to reconstruct that physical reality.
The full paper is available: doi.org/10.1103/3njc-5tlx
