  The convergence of stochastic interacting particle systems in the mean-field
limit to solutions of conservative stochastic partial differential equations is
established, with optimal rate of convergence. As a second main result, a
quantitative central limit theorem for such SPDEs is derived, again, with
optimal rate of convergence.
  The results apply, in particular, to the convergence in the mean-field
scaling of stochastic gradient descent dynamics in overparametrized, shallow
neural networks to solutions of SPDEs. It is shown that the inclusion of
fluctuations in the limiting SPDE improves the rate of convergence, and retains
information about the fluctuations of stochastic gradient descent in the
continuum limit.
