The Paulsen problem is a basic open problem in operator theory.  We define a continuous version of the operator scaling algorithm to solve this problem.  A key step is to show that the continuous operator scaling algorithm converges faster in a perturbed input. To this end, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.  The talk will be self-contained.   Joint work with Tsz Chiu Kwok, Yin Tat Lee, and Akshay Ramachandran.