Teaching a Neural Network to Predict the Birth of the First Stars

Understanding how the very first stars, called Population III stars, formed is one of the big questions in astronomy. These stars were made of the simplest ingredients, just hydrogen and helium from the Big Bang, without the heavier elements we see in later stars like our Sun. To study their formation, scientists need to simulate not only how gravity pulls gas together, but also how temperature and chemical reactions change during the process. This is very computationally expensive, since following all the reactions requires solving complicated equations step by step. In their paper, Sojun Ono and Kazuyuki Sugimura present a new approach: using neural networks to act as “emulators” that can predict these chemical and thermal changes much faster than traditional methods.

Why Neural Networks?

Traditional simulations struggle because the equations are “stiff,” meaning they require many small steps to compute accurately. This becomes especially hard when simulating the collapse of a gas cloud over an enormous range of densities, from extremely diffuse gas to the dense centers where protostars form. Neural networks, once trained, can make these predictions much more quickly. The authors build on earlier machine learning attempts, but those older methods only worked in limited conditions and could not handle the wide range of environments needed for star formation studies.

Methods: Breaking Down the Problem

To tackle this, Ono and Sugimura used a neural network framework called DeepONet, which is designed to learn how equations behave over time. Instead of training one network across the entire range of densities (from 10⁻³ to 10¹⁸ particles per cubic centimeter), they split the problem into five smaller ranges and trained a separate network for each. This helps the models stay accurate across 21 orders of magnitude in density. They also introduced a second improvement: a timescale-based update method. This technique prevents the neural network from drifting into unphysical results when used repeatedly in small time steps, a common problem in simulations.

Results: Accuracy and Speed

The authors tested their emulator on both individual snapshots and full simulations of collapsing gas. The neural networks were able to reproduce the results of traditional methods with errors below 10% in most cases, except for one rare chemical species that is usually negligible anyway. The emulator was not only accurate but also much faster, about ten times faster on CPUs and more than a thousand times faster on GPUs. This makes it particularly promising for large 3D simulations of star formation, where speed is critical.

Fixing the Short-Timestep Problem

When simulations use very small time steps, standard neural networks tend to accumulate errors that ruin the results. Ono and Sugimura’s timescale-based method solves this issue by scaling the predictions according to how quickly variables are expected to change physically. In practice, this means their emulator can remain stable and accurate even when used many thousands of times in a row, making it robust enough for real-world applications.

Discussion and Outlook

The study demonstrates that machine learning can be used to model the chemical and thermal evolution of primordial gas across the full range of densities relevant to Population III star formation. While this paper focused on a simple case without metals or radiation, the framework could be extended to more complex conditions in the future. This could dramatically speed up our ability to simulate early star formation and perhaps answer lingering questions about the first stars in the Universe.

Source: Ono