Partial differential equations (PDE) that arise from physics are usually multiscale in nature. These PDEs include small parameters that differ significantly from the scales at which the equations are typically considered, making it difficult to solve. Traditionally, solving such multiscale systems requires a careful integration of analytical insights into numerical solvers. However, modern multiscale PDEs are usually nonlinear, complex and high-dimensional, making analytical characterization challenging and often leading to the failure of classical numerical solvers. With the increasing ability to collect and process massive volumes of data, a pertinent question arises: can data be leveraged to solve these multiscale problems? This dissertation aims to explore this possibility for elliptic and wave-type equations and their inverse problems. For nonlinear elliptic equations, we investigate how data can be used to enhance the efficiency of multiscale PDE solvers. Specifically, we propose a domain decomposition framework that makes use of the compressibility of solution manifolds and incorporates two strategies from data science: neural networks and manifold learning. We demonstrate the effectiveness of our numerical method across various nonlinear elliptic PDEs. For wave-type equations, we explore how to select appropriate measured data for solving multiscale inverse problems. We propose two formulations of inverse scattering problems with new data collection processes in both time-dependent and time-independent settings, inheriting the well-posedness of the Liouville inverse scattering problem in the high-frequency limit.