Dynamic Heuristic Optimization in High-Order Runge–Kutta Schemes Using Reinforcement Learning and Genetic Algorithms

This paper introduces a hybrid optimisation framework that integrates Genetic Algorithms (GAs) and Reinforcement Learning (RL) for the construction of high-order Runge–Kutta (RK) schemes. Such schemes underpin accurate and efficient time integration in computational science and engineering, with applications ranging from fluid dynamics and chemical kinetics to orbital mechanics and neural ODEs. Traditional optimisation of RK coefficients with interior-point solvers becomes increasingly infeasible as stage counts grow, due to the rapid increase in parameters and the highly nonconvex nature of the feasible region. Our approach addresses this by introducing dynamic algebraic heuristics that contract the search space through symbolic relations among coefficients. GA provides global exploration via mutation, while RL adaptively refines candidate heuristics based on reward signals, guiding optimisation towards stable regions while strictly enforcing order conditions. Empirical studies on third-order Extended-Stability Runge–Kutta (ESRK) schemes show reductions in Interior point optimiser (IPOPT) iteration counts of up to 36.7% for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i>-coefficients, with additional improvements of 32.5% and 24.5% for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i>- and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i>-coefficients. Statistical validation using t-tests and ANOVA confirms the significance of these improvements. Symbolic verification demonstrates that the discovered heuristics generalise across stage counts, maintaining full rank of the order-condition Jacobians and preserving third-order accuracy. Benchmark tests on Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE) systems, including the Brusselator, confirm both expected convergence and efficiency gains. These results establish GA–RL-guided optimisation as a scalable, robust methodology for designing advanced RK integrators, with potential to extend stability-optimised schemes to both classical simulations and modern machine learning contexts.