Learning Fast Approximate Solvers for PDE-Constrained Optimization

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 are tightly coupled with the PDE state variables, and the feasible set is implicitly defined by the governing PDE constraints. This coupling makes these problems computationally demanding, as conventional numerical methods discretize the governing PDE into high-dimensional algebraic systems that must be repeatedly solved throughout the optimization process. To address these challenges, this paper introduces a novel learning-based framework to approximate solutions to PDE-constrained optimization tasks in near real time. The proposed method integrates a dynamic predictor with an optimization surrogate. The dynamic predictor, a time-discrete Neural Operator Lu_2021, efficiently approximates system trajectories governed by PDE dynamics, while the optimization surrogate leverages proxy optimizer techniques kotary2021end 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 including Burgers' equation, heat equation, and voltage regulation, and demonstrate that it achieves solution quality comparable to state-of-the-art control-based algorithms such as the Direct Method and Model Predictive Control, while achieving computational speedups of up to four orders of magnitude.