Machine Learning-Based Adaptive Step Size Control for Numerical Solution of Ordinary Differential Equations

Abstract

Adaptive step-size control plays a key role in the accurate and efficient numerical solution of ordinary differential equations (ODEs). This work introduces a machine learning–enhanced Runge–Kutta method (ML-RK45) that employs a learned predictive policy for step-size selection, moving beyond conventional error-based adaptive strategies. The proposed approach is evaluated against fixed-step fourth-order Runge–Kutta (RK4) and the classical adaptive Runge–Kutta 4(5) method (RK45) using four benchmark problems: exponential decay, harmonic oscillation, nonlinear autonomous systems, and stiff linear equations. Results show that ML-RK45 significantly reduces the number of solver steps and function evaluations while maintaining solution accuracy within prescribed tolerances, albeit with increased computational overhead in CPU time due to neural-network inference. In stiff and oscillatory problems, step-count reductions lead to improved algorithmic efficiency and enhanced numerical stability. Iteration counts decrease by up to 65% for exponential decay and 89% for stiff systems, while oscillatory problems exhibit substantially reduced phase and amplitude errors. Overall, the results demonstrate the potential of predictive machine learning strategies to enhance adaptive ODE solvers, particularly in applications where stability and step efficiency are critical.