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.

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