Symbolic Regression for Interpretable Emulation of Proton Collective Flow in Intermediate-Energy Heavy-Ion Collisions

In the heart of matter, where protons and neutrons bind together to form the nuclei of atoms, lies a world governed by forces that are difficult to measure directly. When scientists smash heavy atomic nuclei together at high speeds, these particles create a fleeting, super-dense state of matter that mimics conditions found in the cores of neutron stars or the early universe. To understand what happens in these collisions, researchers rely on complex computer simulations that act as virtual laboratories. These models calculate how particles interact based on specific rules about nuclear forces and the stiffness of matter. However, running these simulations is incredibly slow and computationally expensive, making it difficult to test every possible variation of the rules to see which one matches reality. Scientists need a way to quickly predict the outcome of these collisions without waiting hours or days for a full computer calculation, and they also need to work backward: to look at the debris from a collision and figure out what the underlying rules must have been.

A team of researchers at East Texas A&M University has developed a new method to solve this problem, offering a faster and more transparent way to navigate the physics of heavy-ion collisions. They focused on two key measurements from these crashes: the way protons flow sideways and the way they stretch into an oval shape as they fly apart. These flow patterns act as fingerprints of the nuclear matter created in the collision. The team used a technique called symbolic regression, which is a form of machine learning designed not just to find patterns, but to write down the actual mathematical formulas that describe them. Unlike standard artificial intelligence models that act like black boxes—taking inputs and giving outputs without revealing how they got there—this new approach produces clear, written equations that anyone can read and understand. The researchers compared this method against deep neural networks, the more common type of machine learning used in physics, to see which tool was better at emulating the slow, heavy simulations.

The study found that the symbolic regression method was just as accurate as the deep neural networks at predicting the flow of protons from the collision conditions. More importantly, it provided the explicit formulas that link the collision settings to the results. Once these formulas were found, they could be used to make predictions millions of times faster than the neural networks. While the neural networks were quicker to set up initially, the symbolic formulas were so efficient that they could run through a million calculations in just a few seconds, whereas the neural networks took several minutes to do the same job. This speed is crucial for scientists who need to run massive statistical tests to understand the uncertainty in their measurements. The researchers also turned the process around, using the formulas to work backward from the observed flow to predict the properties of the nuclear matter, such as how much the particles resist being squeezed together.

However, the researchers discovered that working backward was not always straightforward. While the method successfully predicted one key property related to how particles scatter off each other, it struggled more with predicting the stiffness of the nuclear matter, especially at lower collision energies. The formulas sometimes produced results that varied more from one run to another compared to the neural networks, suggesting that the relationship between the flow and the stiffness of matter is complex and perhaps not fully captured by the simple formulas they allowed the computer to find. The team noted that the flow data alone might not contain enough information to pin down the stiffness perfectly, or that the mathematical forms they used were too simple to describe the subtle wiggles and ripples in the data. Despite these challenges, the ability to generate clear, readable equations that run at lightning speed offers a powerful new tool. It allows physicists to see exactly how different factors influence the outcome of a collision, rather than just trusting a computer's guess.

The researchers tested their approach using data from real experiments conducted by the FOPI and HADES collaborations, which have been studying these collisions for years. They found that for the most part, their new formulas could reproduce the results of the heavy computer simulations with high precision. The method worked well across a wide range of collision speeds, from 150 to 800 million electron volts per nucleon. The team also checked how sensitive their results were to the amount of data used for training and testing, finding that the accuracy remained stable even when they changed the size of the data sets. They also confirmed that the method worked equally well whether they used the raw numbers from the simulations or numbers that had been scaled down to make them easier to handle, though they chose to keep the original numbers to ensure the final formulas made physical sense.

Ultimately, this work demonstrates that machine learning can do more than just make predictions; it can help scientists discover the underlying language of nature. By providing explicit equations, symbolic regression bridges the gap between complex computer models and human understanding. While the neural networks remain a strong tool for consistency, the new method offers a unique advantage: it gives researchers a fast, transparent map of the physics at play. This is particularly valuable for future studies where scientists need to explore vast possibilities to understand the fundamental properties of matter. The study suggests that while some relationships in nuclear physics are too complex for simple formulas, many others can be captured clearly and quickly, opening the door to deeper insights into the building blocks of the universe without the need for endless, slow computer runs.