A model reduction method based on nonlinear optimization for multiscale stochastic optimal control problems

Abstract

This paper proposes a non-intrusive, data-driven reduced-order modeling framework for stochastic optimal control problems governed by partial differential equations. The control problem is formulated with a quadratic cost functional and stochastic PDE constraints, and an L2-optimal reduced-order model is constructed to directly approximate the parameter-to-output mapping. The model is obtained by minimizing the L2 norm of the output error via gradient-based optimization, requiring only input-output data without access to the full-order system matrices or state variables. To efficiently generate high-fidelity training data for multiscale problems, the Generalized Multiscale Finite Element Method (GMsFEM) is employed as an offline solver. The proposed framework ensures accuracy in control-relevant outputs while maintaining computational complexity independent of the original PDE dimension, making it suitable for real-time applications. Numerical experiments on stochastic diffusion and advection-diffusion equations demonstrate the accuracy, efficiency, and robustness of the method.