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In this work we derive the rate of convergence for the empirical measure of amoderately interacting stochastic particle system towards the Keller-Segel andBurgers equations. In the case of the Keller-Segel equation on a$d$-dimensional torus, we obtain almost sure convergence of the mollifiedempirical measure to the solution of the PDE in some $L^1\cap L^p$ spaces witha rate of order $N^{-\frac{1}{2d+1}}$. The same holds for the genuine empiricalmeasure in Wasserstein distance. The result holds in both subcritical andsupercritical cases for the sensitivity parameter $\chi$ of the equation up tothe maximal existence time of the PDE. This extends our previous work tomoderately interacting particle systems without cutoff in the drift term. Then,we extend our method to Burgers equation in $\mathbb{R}$ and obtain a rate ofconvergence of the order $N^{-1/8}$.

Quantitative approximation of the Keller-Segel and Burgers equations by moderately interacting particles