Reinforced Data-Driven Estimation for Spectral Properties of Koopman Semigroup in Stochastic Dynamical Systems

Analyzing the spectral properties of the Koopman operator is crucial for understanding and predicting the behavior of complex stochastic dynamical systems. However, the accuracy of data-driven estimation methods, such as Extended Dynamic Mode Decomposition (EDMD) and its variants, are heavily dependent on the quality and location of the sampled trajectory data. This paper introduces a novel framework, Reinforced Stochastic Dynamic Mode Decomposition, which integrates Reinforcement Learning (RL) with Stochastic Dynamic Mode Decomposition (SDMD) to automatically guide the data collection process in stochastic dynamical systems. We frame the optimal sampling strategy as an RL problem, where an agent learns a policy to select trajectory initial conditions. The agent is guided by a reward signal based on \emph{spectral consistency}, that is a measure of how well the estimated Koopman eigenpairs describe the system's evolution balanced with an exploration bonus to ensure comprehensive coverage of the state space. We demonstrate the effectiveness of our approach using Bandit algorithm, Deep Q-Network (DQN), and Proximal Policy Optimization (PPO) algorithms on canonical systems including the double-well potential, the stochastic Duffing oscillator and the FitzHugh-Nagumo model. Our results show that the RL agent automatically discovers dynamically significant regions without any prior knowledge of the system. Rigorous theoretical analysis establishes convergence guarantees for the proposed algorithms, directly linking the final estimation accuracy to the quality of the learned sampling policy. Our work presents a robust, automated methodology for the efficient spectral analysis of complex stochastic systems.