Neural dynamic surface control (NDSC) is an effective technique for the tracking control of nonlinear systems. The objective of this article is to improve closed-loop transient performance and reduce the number of learning parameters for a strict-feedback nonlinear system with unknown control gains. For this purpose, a predictor-based NDSC (PNDSC) approach is presented. It introduces Nussbaum functions and predictors into the traditional NDSC for nonlinear systems with unknown control gains. Unlike NDSC that uses surface errors to update the learning parameters of neural networks (NNs), the PNDSC employs prediction errors for the same purpose, leading to improved transient performance of closed-loop control systems. To reduce the number of learning parameters, the PNDSC is further embedded with the technique of the minimal number of learning parameters (MNLPs). This avoids the problem of the "explosion of learning parameters" as the order of the system increases. A Lyapunov-based stability analysis shows that all signals are bounded in the closed-loop systems under PNDSC embedded with MNLPs. Simulations are conducted to demonstrate the effectiveness of the PNDSC approach presented in this article.