System identification methods are primarily divided into two categories: one is based on first-principles modeling, and the other on data-driven modeling via machine learning. Although data-driven models provide higher accuracy, their lack of physical interpretability can lead to challenges in validating model reliability, thereby limiting their widespread application in engineering. As a novel data-driven modeling approach, the Elementary Mechanical Network (EMN) adheres to the existing mechanical theory framework, ensuring that the identified results can be interpreted from a mechanical perspective. However, due to the numerous constraints within the EMN structure, its modeling accuracy is inferior to other data-driven methods such as neural networks. Therefore, enhancing the network's fitting capability within the existing model architecture is key to further development and application of EMN. This paper first develops a set of differential-algebraic explicit solution frameworks for EMN from the perspective of numerical computation and designs numerical solving algorithms, including the Euler method, the second-order Runge-Kutta method, and the fourth-order Runge-Kutta method based on this framework. Next, numerical examples are provided to analyze the computational accuracy and initial sensitivity of EMN under the new framework, while comparing the three numerical methods in terms of solving capability, stability, and time complexity, offering a basis for subsequent method selection. Finally, simulation experiments are conducted to build an equivalent model of LuGre friction by training the EMN. The experimental results show that the trained EMN achieves a mean square error (MSE) of only 0.0018 and can effectively reproduce the internal state variables of the model, verifying the feasibility of EMN for both quantitative and qualitative feature approximation.