用神经网络加速宇宙早期粒子演化模拟，提升计算效率与精度。
Neural Boltzmann Equations
- 用神经分布函数编码粒子属性，支持快速参数扫描。
- 结合蒙特卡洛重要性采样，高效计算高维相空间积分。
- 采用自然梯度法演化系统，适合复杂物理过程研究。
早期宇宙中粒子的演化由玻尔兹曼方程描述，涉及高维相空间积分。传统方法使用固定动量网格上的数值积分，计算代价随系统复杂度急剧上升，严重限制了可研究过程的复杂性。本文提出神经玻尔兹曼方程（NBE），融合三个关键思想：第一，粒子性质由物理启发的神经分布函数表示，其参数可通过神经网络预测，实现高效参数扫描；第二，相空间积分采用蒙特卡洛方法，并引入对撞机物理中的重要性采样技术；第三，使用自然梯度法演化系统。在验证各组件优势后，利用该框架实现了对早期宇宙有效相对论性中微子自由度数的高精度计算。
原文摘要 · Abstract (English)
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.
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