发表机构
Institute for Particle Physics Phenomenology; Durham University; University of Liverpool（粒子物理现象学研究所; 杜伦大学; 利物浦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对经典玻尔兹曼方程方法扩展性差的问题，提出神经玻尔兹曼方程（NBEs），结合神经分布函数、蒙特卡洛积分与自然梯度方法，实现高效参数扫描并完成中微子自由度的精确计算。
AI 中文摘要
早期宇宙中粒子的动力学由玻尔兹曼方程描述，该方程涉及高维相空间积分。经典方法采用正交积分并在固定动量网格上演化系统，其在复杂系统和参数扫描时扩展性极差，严重限制了可研究过程的复杂度。我们提出神经玻尔兹曼方程（Neural Boltzmann Equations, NBEs），结合三个耦合概念克服这些局限：第一，粒子特性由受物理启发的神经分布函数编码，其参数可通过神经网络预测，实现高效参数扫描；第二，利用对撞机物理中的重要性采样工具，通过蒙特卡洛方法计算相空间积分；第三，采用自然梯度方法演化系统。在展示NBEs的各方面优势后，我们使用该框架对早期宇宙中相对论性中微子自由度的有效数量进行了精确计算。
英文摘要
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 the most numerically precise calculation to date of the effective number of relativistic neutrino degrees of freedom, $N_\mathrm{eff}$, in the Standard Model.
Commentsv1: 21 pages, 4 figures, 1 table. v2: 30 pages, 7 figures, 6 tables, submitted version with extended experiments
