We introduce our two recent efforts to implement GPU-accelerated solvers; one is ExaHyPar for hyperbolic-parabolic partial differential equations with source terms on Cartesian grids and the other ExaTron for nonlinear optimization. Starting with basic concepts and methodology for GPU computing, we present computational challenges we have encountered during our development and describe our design principles based on numerical justification to efficiently address those challenges. Our basic approach for PDE solves is to perform all the grid-point computations on GPUs. In the case of nonlinear optimization, we employ a batch nonlinear programming scheme that decomposes a given large problem into many small subproblems and solve them in parallel on GPUs. A key feature of our solvers is that they operate fully on GPUs without requiring data transfers between host and device. This feature enables us to achieve significantly accelerated computations. We demonstrate computational performance of ExaTron on large alternating current optimal power flow problems on the Summit supercomputers using multiple GPUs. We conclude this talk with briefly introducing our ongoing work that targets to integrate deep neural networks with our solvers.