Second order Ordinary Differential Equations (ODE) were considered. Numerov-like techniques employing effectively seven stages per step and sharing eighth algebraic order were under examination for numerically solving them. The coefficients of these methods were contingent on four independent parameters. To tackle issues with oscillatory solutions, we typically aimed to fulfill specific criteria such as minimizing phase-lag, expanding the periodicity interval, or even neutralizing amplification errors. These latter attributes stemmed from a test problem mimicking an ideal trigonometric trajectory. Here, we suggested training the coefficients of the chosen method family across a broad spectrum of pertinent problems. Following this training using the differential evolution method, we identified a particular method that surpassed others in this category across an even broader array of oscillatory problems.

Runge–Kutta (RK) pairs remain among the most effective tools for the numerical integration of ordinary differential equations; however in the scalar autonomous setting, their structure allows further efficiency. In this case, the system of order conditions simplifies, with fewer equations needing to be satisfied, which in turn enables the construction of embedded pairs of higher accuracy than in the general case. Notably, one may design pairs of order 7(5) requiring only eight stages per step, whereas conventional RK pairs with the same number of stages are limited to order 6(5). In the present work, we derived the modified set of equations of condition up to seventh order, and by means of differential evolution techniques, constructed a new embedded pair of order 7(5) specifically adapted to scalar autonomous problems. The performance of the method was assessed in the context of system identification through neural ODEs, where it was used to approximate governing dynamics from data. The logistic growth and saturating cubic models were employed as a representative test cases, illustrating both the efficiency advantages of the proposed scheme. Numerical experiments confirmed that the new pair provides a valuable bridge between high–order RK methodology and modern machine learning approaches to dynamical systems.