Model-Based Closed-Loop Control Algorithm for Stochastic Partial Differential Equation Control

Abstract

Neural operators have demonstrated promise in modeling and controlling systems governed by Partial Differential Equations (PDEs). Beyond PDEs, Stochastic Partial Differential Equations (SPDEs) play a critical role in modeling systems influenced by randomness, with applications in finance, physics, and beyond. However, controlling SPDE-governed systems remains a significant challenge. On the one hand, the regularity of the system's state (which can be intuitively understood as smoothness) deteriorates, making modeling and generalization more challenging. On the other hand, this stochasticity also renders control more unstable and thus less accurate. To address this gap, we propose the Model-Based Closed-Loop Control Algorithm (MB-CC), the first model-based closed-loop control method for SPDEs. MB-CC introduces two key innovations to enhance control robustness and efficiency: a Regularity Feature (RF) block and a closed-loop strategy with an operator-encoded policy network. The RF block, inspired by the regularity structure theory of SPDEs, addresses noise-induced irregularities by transforming the network's input—including the system state and noise-perturbed external forces—into a refined feature space for improved forward prediction. Compared to previous works using regularity features, we introduce a new parameterization, data augmentation, and extend the RF block as a plug-and-play component. Additionally, to achieve closed-loop control, we introduce an operator-encoded policy network to map the current state to optimal control, which integrates physical priors and swiftly makes decisions based on states returned by the environment. We conduct a systematic evaluation of MB-CC on two notable SPDEs, showcasing its effectiveness and efficiency. The ablation studies show its ability to handle stochasticity more effectively.