Partial differential equation (PDE)-constrained optimization arises in many scientific and engineering domains, such as energy systems, fluid dynamics and material design. In these problems, the decision variables (e.g., control inputs or design parameters) are tightly coupled with the PDE state variables, and the feasible set is implicitly defined by the governing PDE constraints. This coupling makes the problems computationally demanding, as it requires handling high dimensional discretization and dynamic constraints. To address these challenges, this paper introduces a learning-based framework that integrates a dynamic predictor with an optimization surrogate. The dynamic predictor, a novel time-discrete Neural Operator (Lu et al.), efficiently approximate system trajectories governed by PDE dynamics, while the optimization surrogate leverages proxy optimizer techniques (Kotary et al.) to approximate the associated optimal decisions. This dual-network design enables real-time approximation of optimal strategies while explicitly capturing the coupling between decisions and PDE dynamics. We validate the proposed approach on benchmark PDE-constrained optimization tasks inlacing Burgers' equation, heat equation and voltage regulation, and demonstrate that it achieves solution quality comparable to classical control-based algorithms, such as the Direct Method and Model Predictive Control (MPC), while providing up to four orders of magnitude improvement in computational speed.

翻译：偏微分方程（PDE）约束的优化问题广泛出现在能源系统、流体动力学和材料设计等诸多科学与工程领域。在这类问题中，决策变量（例如控制输入或设计参数）与PDE状态变量紧密耦合，可行集由主导的PDE约束隐式定义。这种耦合使得问题计算量巨大，因为它需要处理高维离散化和动态约束。为应对这些挑战，本文提出了一种基于学习的框架，该框架将动态预测器与优化代理模型相结合。动态预测器是一种新颖的时间离散神经算子（Lu等人提出），能够高效近似由PDE动力学支配的系统轨迹；而优化代理模型则利用代理优化器技术（Kotary等人提出）来近似相关的优化决策。这种双网络设计能够在显式捕捉决策与PDE动力学之间耦合关系的同时，实现对最优策略的实时近似。我们在包括Burgers方程、热传导方程和电压调节在内的基准PDE约束优化任务上验证了所提方法，结果表明其求解质量可与基于经典控制的方法（如直接法和模型预测控制（MPC））相媲美，同时计算速度提升了高达四个数量级。