The problem of model reduction for complex systems is an active area of research. In this talk I want to discuss how the physics inspired concepts of scale dependence and renormalization can be used to facilitate the construction of accurate reduced models for complex problems. In particular, I will be presenting the application of these concepts to the problem of detecting and tracking singularities of time-dependent partial differential equations. Results for the inviscid Burgers, the critical nonlinear Schrödinger and the incompressible Euler equations will be used to illustrate the constructions.