Obtaining compressed representations of entangled systems is an important open problem in quantum mechanics. The Density Matrix Renormalization Group (DMRG) algorithm introduced by S. R. White in 1992  has been successful at solving one-dimensional cases but does not generalize well to arbitrary dimensions. We explore the possibility of using neural network models to solve ground state problems in place of DMRG. In experiments on a system of four spin-1 particles interacting by the 1D Heisenberg Hamiltonian, we show that this approach can approximate ground state energies and Matrix Product State coefficients to a mean percent error of less than 5%. Furthermore, we use deep learning to obtain MPS coefficients for low-lying energy states directly from the system Hamiltonian. Our findings suggest that neural networks, which generalize well to arbitrary di- mensions, could be useful tools for solving 2D and 3D systems where DMRG fails.