用遗传算法与强化学习优化高阶龙格库塔法，提升计算效率25%。
Optimising 4th-Order Runge-Kutta Methods: A Dynamic Heuristic Approach for Efficiency and Low Storage
- 结合遗传算法探索搜索空间，用强化学习动态优化启发式规则。
- 在典型方程上实现25%的求解时间减少，保持四阶精度和稳定性。
- 适合需要高效低存储数值模拟的科研与工程领域，如流体动力学。
扩展稳定龙格库塔（ESRK）方法在科学与工程的大规模计算中至关重要，涵盖天气预报、气动分析及复杂生物建模。然而，高阶低存储格式在精度、稳定性与计算效率之间仍难平衡。本文提出一种混合遗传算法（GA）与强化学习（RL）的启发式自动发现方法，用于优化低存储ESRK格式。不同于传统依赖人工设计或穷举搜索的方法，该框架利用GA驱动的变异进行搜索空间探索，并引入类似强化学习的状态转移机制，动态优化启发式选择，实现参数系统性缩减，在保持四阶精度的同时显著提升计算效率。在1D与2D布鲁塞尔атор系统及稳态纳维-斯托克斯方程等基准测试中验证，最优启发式使IPOPT求解时间减少25%，同时维持数值稳定性和准确性。结果表明，自适应启发式发现可有效提升高保真模拟的资源效率，拓展低存储龙格库塔方法在实际计算流体力学、物理仿真等领域的应用。本研究建立了数值方法启发式优化的新范式，为深度强化学习与自动化机器学习驱动的启发式搜索开辟路径。
原文摘要 · Abstract (English)
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
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