Conventional numerical methods for dynamic problems can be computationally intensive when addressing localized features that require local adaptive refinement. Local adaptive refinement is tedious to meet the regularity requirements, and analytical enrichment functions that capture local features are often unavailable. These complexities become more pronounced in transient problems. This work introduces a neural network-enriched Reproducing Kernel Particle Method (NN-RKPM) for solving dynamic problems based on action minimization under a symplectic space-time framework [1]. In this approach, we employ RKPM as the background spatial discretization, which we then adaptively enrich by introducing extrinsic neural network basis functions. The neural network enrichment functions are pretrained during the offline stage to learn specific local features through Ritz-type energy minimization, with proper orthogonality constraints applied to ensure the optimization enriches the local strain solution. With NN-RKPM, the dynamic problem is an optimization problem, with time-domain interpolation used to assemble and discretize the NN-enriched action functional. The principle of least action drives the evolution of the background RKPM and NN-enriched time-dependent solution through variational integration algorithms derived from first principles. The proposed method offers a unified framework consistent with classical field theory and variational integrators, and accurately captures the time-dependent response of mechanical systems. This application of mechanics and machine learning is then evaluated for computational efficiency and solution accuracy using several local-feature-specific test cases.