用强化学习加速求解二维柯尔莫哥洛夫-希瓦辛斯基方程的不动点。
Numerical solutions of fixed points in two-dimensional Kuramoto-Sivashinsky equation expedited by reinforcement learning
- 用深度强化学习优化牛顿-克里洛夫法的初始猜测。
- 发现文献未报道的新不动点解，收敛速度提升显著。
- 适合研究复杂动力系统与控制优化的科研人员。
本文提出一种结合深度强化学习（DRL）与无雅可比牛顿-克里洛夫（JFNK）方法的新策略，用于高效求解二维柯尔莫哥洛夫-希瓦辛斯基方程（2D KSE）中的不动点。传统JFNK方法依赖良好初始猜测以实现快速收敛，本文通过设计合理的奖励函数，利用DRL作为预处理步骤，显著改善初始解质量。实验揭示了文献中尚未报道的新不动点解，并基于并行强化学习技术探索了在已知不动点间调控系统轨迹的最优控制策略。该联合方法显著提升了在2D KSE中寻找新不动点解的效率，对高维动力系统的研究具有借鉴意义。
原文摘要 · Abstract (English)
This paper presents a combined approach to enhancing the effectiveness of Jacobian-Free Newton-Krylov (JFNK) method by deep reinforcement learning (DRL) in identifying fixed points within the 2D Kuramoto-Sivashinsky Equation (KSE). JFNK approach entails a good initial guess for improved convergence when searching for fixed points. With a properly defined reward function, we utilise DRL as a preliminary step to enhance the initial guess in the converging process. We report new results of fixed points in the 2D KSE which have not been reported in the literature. Additionally, we explored control optimization for the 2D KSE to navigate the system trajectories between known fixed points, based on parallel reinforcement learning techniques. This combined method underscores the improved JFNK approach to finding new fixed-point solutions within the context of 2D KSE, which may be instructive for other high-dimensional dynamical systems.
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