Sparsity plays a central role in recent developments of many fields such as signal and image processing, compressed sensing, statistics, and optimization. In practice, sparsity is promoted through the additional of an L1 norm (or related quantity) as a constraint or penalty in a variational model. Motivated by the success of sparsity-inducing methods in imaging and information sciences, there is a growing interest in exploiting sparsity in dynamical systems and partial differential equations. In this talk, we will investigate the connections between compressed sensing, sparse optimization, and numerical methods for nonlinear differential equations. In particular, we will discuss about sparse modeling as well as the advantage of sparse optimization in solving various differential equations arising from physical and data sciences.