Numerical solutions of fixed points in two-dimensional Kuramoto-Sivashinsky equation expedited by reinforcement learning
This incremental improvement addresses computational challenges in high-dimensional dynamical systems for researchers in applied mathematics and physics.
The paper tackles the problem of finding fixed points in the 2D Kuramoto-Sivashinsky equation by enhancing the Jacobian-Free Newton-Krylov method with deep reinforcement learning to improve initial guesses, resulting in the discovery of new fixed points not previously reported in the literature.
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.