This work introduces a novel neural network-enriched reproducing kernel particle method (NN-RKPM) approach for solving static and dynamic problems via artificial neural networks with an energy-based loss function minimization [1]. The flexibility and adaptivity of the NN function space are utilized to capture complex solution patterns that the conventional Galerkin methods fail to capture. The NN enrichment is constructed by combining pre-trained feature-encoded NN blocks with additional untrained NN blocks. The pre-trained NN blocks learn specific local features during the offline stage, enabling efficient enrichment of the approximation space during the online stage through the energy minimization. The NN enrichment is introduced under the Partition of Unity (PU) framework, ensuring convergence of the proposed method. The proposed NN-PU approximation and feature-encoded transfer learning form an adaptive approximation framework, termed the neural-refinement (n-refinement). This work is then extended to a novel Iterative Action Minimization Method (IAMM) for solving dynamic problems. The framework employs reproducing kernel (RK) approximations on a coarse background domain, enriched by neural network functions, with a temporal discretization compatible with the action formulation. The evolution of the background RK solutions and NN enrichment functions is driven by minimizing the action functional in accordance with the principle of minimal action. Unlike traditional energy methods like the Ritz method, which minimize total potential energy in a quasi-static sense to locate the ground state, action minimization addresses the time-dependent and path-dependent nature of dynamic systems, incorporating the potential and kinetic fields within a symplectic formulation of mechanics for a Newtonian four-space. This approach offers a unified and mathematically elegant framework consistent with classical field theory and has been shown to accurately capture the time-dependent evolution of mechanical systems. REFERENCES [1] Baek, J., Wang, Y., & Chen, J. S. (2024). N-adaptive Ritz method: A neural network enriched partition of unity for boundary value problems. Computer Methods in Applied Mechanics and Engineering, 428, 117070.