We demonstrate digital predistortion (DPD) using a novel, neural-network (NN) method to combat the nonlinearities in power amplifiers (PAs), which limit the power efficiency of mobile devices, increase the error vector magnitude, and cause inadequate spectral containment. DPD is commonly done with polynomial-based methods that use an indirect-learning architecture (ILA) which can be computationally intensive, especially for mobile devices, and overly sensitive to noise. Our approach using NNs avoids the problems associated with ILAs by first training a NN to model the PA then training a predistorter by backpropagating through the PA NN model. The NN DPD effectively learns the unique PA distortions, which may not easily fit a polynomial-based model, and hence may offer a favorable tradeoff between computation overhead and DPD performance. We demonstrate the performance of our NN method using two different power amplifier systems and investigate the complexity tradeoffs.