Optimising 4th-Order Runge-Kutta Methods: A Dynamic Heuristic Approach for Efficiency and Low Storage

Extended Stability Runge-Kutta (ESRK) methods are crucial for solving large-scale computational problems in science and engineering, including weather forecasting, aerodynamic analysis, and complex biological modelling. However, balancing accuracy, stability, and computational efficiency remains challenging, particularly for high-order, low-storage schemes. This study introduces a hybrid Genetic Algorithm (GA) and Reinforcement Learning (RL) approach for automated heuristic discovery, optimising low-storage ESRK methods. Unlike traditional approaches that rely on manually designed heuristics or exhaustive numerical searches, our method leverages GA-driven mutations for search-space exploration and an RL-inspired state transition mechanism to refine heuristic selection dynamically. This enables systematic parameter reduction, preserving fourth-order accuracy while significantly improving computational efficiency.The proposed GA-RL heuristic optimisation framework is validated through rigorous testing on benchmark problems, including the 1D and 2D Brusselator systems and the steady-state Navier-Stokes equations. The best-performing heuristic achieves a 25\% reduction in IPOPT runtime compared to traditional ESRK optimisation processes while maintaining numerical stability and accuracy. These findings demonstrate the potential of adaptive heuristic discovery to improve resource efficiency in high-fidelity simulations and broaden the applicability of low-storage Runge-Kutta methods in real-world computational fluid dynamics, physics simulations, and other demanding fields. This work establishes a new paradigm in heuristic optimisation for numerical methods, opening pathways for further exploration using Deep RL and AutoML-based heuristic search