Title: Learning to Solve Partial Differential Equations Constrained Optimization Problems
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: Partial Differential Equations, Constrained Optimization, Proxy Optimizers, Neural Operators

Abstract: 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 discretizations 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., 2021), efficiently approximates system trajectories governed by PDE dynamics, while the optimization surrogate leverages proxy optimizer techniques (Kotary et al., 2021) 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 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.

TL;DR: We introduce a novel learning-based method to solve partial differential equations constrained optimization problems in real time.