Neural Boltzmann Equations
2026-08-24 • Machine Learning
Machine Learning
AI summaryⓘ
The authors present a new method called Neural Boltzmann Equations (NBEs) to study how particles behaved in the early universe more efficiently. They use neural networks to represent particle properties and predict parameters, combine this with Monte Carlo sampling techniques to handle complex calculations, and apply a special optimization method called the natural gradient to evolve the system over time. This approach helps overcome problems with older methods that were slow and could not handle complicated scenarios well. They demonstrate their method by precisely calculating the effective number of neutrino types influencing the early universe.
Boltzmann equationphase-space integralneural networksMonte Carlo integrationimportance samplingnatural gradientearly universerelativistic neutrinoseffective degrees of freedom
Authors
Jonas Spinner, Jack Shergold
Abstract
The dynamics of particles in the early universe are described by Boltzmann equations, which involve high-dimensional phase-space integrals. Classical approaches use quadrature integration and evolve the system on a fixed momentum grid, which scales poorly to complicated systems and parameter scans, severely limiting the complexity of processes that can be studied. We introduce Neural Boltzmann Equations (NBEs), which combine three coupled concepts to overcome these limitations. First, particle properties are encoded in physics-inspired neural distribution functions, with parameters that can be predicted using neural networks, enabling efficient parameter scans. Second, phase-space integrals are evaluated with Monte Carlo, using importance sampling tools from collider physics. Third, we use the natural gradient method to evolve the system. After demonstrating the individual benefits of NBEs, we use the framework to perform a precision calculation of the effective number of relativistic neutrino degrees of freedom in the early universe.