Regularizing divergent integrals is a key step in the mathematical understanding of quantum field theories via their correlation functions. An analytic notion of regularization was introduced and developed by Li-Zhou that aims to assign finite values to divergent integrals on configuration spaces of Riemann surfaces. This notion provides a satisfying regularization scheme that meets various expectations from two dimensional chiral conformal field theories. In this talk, I will explain an extended notion of regularized integrals, and more importantly provide cohomological formulations using the tools of current cohomology and mixed Hodge structures. I will also explain practical ways of constructing representatives of the corresponding cohomology classes in terms of smooth differential forms.