Runge-Kutta pairs for scalar autonomous Neural ODEs

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.