the essence of the algorithm that brought Hinton and Hopfield the Nobel Prize

The red_mad_robot analytical center has collected for you the main things you need to know about the 2024 Nobel Prize in Physics.

The Nobel Committee awarded the Physics Prize to scientists who used fundamental physical principles to develop machine learning. Thanks to this algorithm, AI will be able to learn and improve independently – without human help. We tell you how closely physics and ML are connected, what the future of neural networks is in matter research, and what the essence of the method that brought Hinton and Hopfield the award is.

Hopfield developed “associative memory”. This method allows you to recover even partially distorted images or other types of data.

Hinton created a neural network based on the ideas of Hopfield and the “Boltzmann machine”. It is able to recognize key elements in images.

Combination of Hopfield network and Boltzmann machine

The Nobel Committee believes that neural networks will find their application in other areas of physics: predicting the properties of molecules and materials, developing solar cells, measuring gravitational waves from the collision of black holes, or, for example, searching for exoplanets.

The influence of physics on machine learning

At first glance, physics and ML may seem like two completely different fields: one is about understanding the fundamental laws of nature, the other is about teaching machines from data. However, physics provided several key concepts that laid the foundation for modern machine learning, such as the design and development of artificial neural networks. Below are examples of how the fundamental concepts in physics that were used in the work of Hopfield and Hinton influenced the development of ML.

Statistical physics

Statistical physics—which, by the way, was used by Hinton—considers systems consisting of many interacting components. It provides a theoretical basis for understanding the behavior of large complex systems such as gases or liquids. The early development of artificial neural networks was largely inspired by this field.

In the 1940s, researchers, including physicists, began to model the brain by mathematically representing neurons and synapses. This early work laid the foundation for the artificial neural networks we use today. The human brain is a complex system of interconnected neurons, and the behavior of such systems can be modeled using equations from statistical physics.

Comparison of the human brain and neural networks

At that time, scientists perceived neural network technology as an analogue of the structure of biological neural connections. And neurobiology came to the aid of mathematical methods with the idea of strengthening the connections of neurons during their interaction: in an artificial neural network, neurons are represented by nodes that have different meanings. These nodes influence each other through connections—conditioned synapses—that can be made stronger or weaker. The network learns, for example, by developing stronger connections between nodes with simultaneously high values.

Energy landscapes

The concept of energy landscapes helps to understand how systems move from one state to another. In machine learning, this analogy is used to describe how an algorithm moves through the space of possible solutions (states) in search of the best solution (the state with the lowest energy).

When training neural networks, researchers often strive to minimize errors, and this process can be thought of as a landscape with hills (high error) and valleys (low error). Algorithms like gradient descent rely on this energy landscape to find the “minimum” (best solution).

A good example of this process is the Hopfield network. It finds the optimal configuration, constantly updating its state until further reduction in energy becomes impossible. This is a reflection of the transition of physical systems to a state with minimal energy.

Chaos theory

Chaos theory shows that even small changes in initial conditions lead to completely different results. It has great implications for predictive modeling in ML.

Chaos theory helps explain why some systems are sensitive to even small fluctuations and are difficult to predict over the long term. In ML, models that deal with chaotic systems, such as weather or stock market forecasts, must account for such unpredictability.

Advanced ML methods such as recurrent neural networks (RNNs), which are similar to Hopfield networks and Hinton networks, and long-term memory models (LSTMs) are particularly well suited for making predictions in chaotic systems because they are designed to deal with sequential, time-dependent data.

John Hopfield: Associative Memory

The Hopfield network uses the physics of material, or more precisely, atomic spin. The network is described in a manner equivalent to describing energy in a spin system and is trained by finding values for connections between nodes, so that the stored images have low energy.

When a Hopfield network receives a distorted or incomplete image, it methodically goes through the nodes and updates their values so that the network's energy drops. Thus, the network works in stages to find the stored image that most closely matches the given one.

Simply put, the network can store and reproduce images and other types of patterns, similar to how the human brain works. It is based on ideas from statistical mechanics, in particular how systems with large numbers of particles can exhibit collective behavior. The network stores patterns in the same way the brain stores associative memories, allowing entire memories to be retrieved from partial or noisy data.

How does the Hopfield network work?

Fully connected neural network with a symmetric connection matrix

Hopfield described a memory model that accesses its contents using an asynchronous parallel processing algorithm. He used a parallel with the physical properties of magnetic materials. They have special characteristics due to their atomic spin, which makes each atom a tiny magnet. The spins of neighboring atoms influence each other, allowing them to spin in the same direction.

The network built by Hopfield consists of nodes with different weights. Each node can store an individual value – either 0 or 1, like pixels in a black and white picture.

The network learns by sequentially calculating weight values to arrive at an equilibrium state. This value is determined by the equivalent physical “energy”, which depends on the weight of all elements in the system. The network is looking for the minimum “energy” at which it “remembers” a certain pattern. If this value decreases when passing a node, then the black pixel becomes white or vice versa. When the goal is achieved, the network reproduces the original image on which it was trained. If the image is slightly distorted and fed to the network input, it will also be restored.

Hopfield's method is special because it allows the network to distinguish between multiple simultaneously stored images. Hopfield compared searching for a stored state online to rolling a ball across a landscape of peaks and valleys, the frictional force of which slows the ball down. If you throw a ball in a certain place, it will roll down to the nearest valley and stop there. If the network is given a pattern close to one of the stored ones, it will continue to move forward in the same way until it ends up at the bottom of a valley in the landscape, thus remembering the closest pattern.

Such a network can be called recurrent. It redirects information back and forth through the layers until the final result is achieved. The Hopfield network is suitable for reconstructing data that contains noise or has been partially erased.

Geoffrey Hinton: a neural network based on the idea of Hopfield and the Boltzmann machine

Geoffrey Hinton decided to use Hopfield's invention as the basis for a new neural network using a different method: Boltzmann machine. This would allow it to autonomously find data properties and identify certain elements in images. To develop the new neural network, Hinton also used methods from statistical physics – the science of systems built from many identical elements.

How does the Hinton neural network work?

The network is trained not on instructions, but on examples that are likely to arise when it is launched. If the same pattern is repeated several times during the learning process, the likelihood of its occurrence becomes even higher.

Training also affects the likelihood of outputting new patterns that are similar to examples from the machine's training. A trained machine can recognize familiar patterns in information it has not previously seen. For example, you meet a friend's brother or sister, and you immediately realize that they must be related. Similarly, a machine can recognize a completely new example if it has encountered a similar category in the training material, and also distinguish it from an example from a different category.

the essence of the algorithm that brought Hinton and Hopfield the Nobel Prize