Much recent progress in data science (e.g., compressed sensing and matrix completion) has come from the use of sparsity and variational principles. This talk is on transfer of these ideas from information science to differential equations and physics. The focus is on variational principles and differential equations whose solutions are spatially sparse; i.e. they have compact support. Analytic results will be presented on the existence of sparse solutions, the size of their support and the completeness of the resulting “compressed modes”. Applications of compressed modes as Wannier modes in density functional theory and for signal fragmentation in radio transmission will be described.