Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computa- tionally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density. A prototypical example that we will consider is the formal identification of a finite system of particles with the singular Dean-Kawasaki equation. We will give a short introduction about the Dean-Kawasaki equation and its applications. Our aim is to introduce a well-behaved nonlinear SPDE that approximates the Dean- Kawasaki equation for a particle system with mean-field interaction both in the drift and the noise term. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation with respect to the particle system using the technique of transport equations on the space of probability measures. This is the joint work with H. Kremp, N. Perkowski and J. Xiaohao. Furthermore, we will discuss possible numerical methods for these problems.
