Learning to Solve Optimization Problems Constrained with Partial Differential Equations

2025-09-29

AI Summary

PDE-constrained optimization—ubiquitous in energy systems, fluid dynamics, and materials design—is hindered by high-dimensional discretization, strong physical coupling, and implicitly defined feasible sets, rendering conventional iterative solvers computationally prohibitive. This paper proposes a dual-network learning framework: a temporal-discrete neural operator explicitly models the PDE state evolution, while a surrogate optimizer directly approximates the optimal decision variables. By jointly learning dynamics and optimization in an end-to-end manner, the method bypasses costly nested iterations and enables real-time inference. On multiple benchmark tasks, it matches the accuracy of direct methods and model predictive control while accelerating inference by up to four orders of magnitude. The key contribution lies in the first integration of neural operators’ physics-informed dynamic modeling with surrogate-based optimization—achieving simultaneous guarantees on physical consistency, generalizability across PDE instances, and computational efficiency.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

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 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.

Problem

• Solving PDE-constrained optimization problems efficiently
• Handling high-dimensional discretization and dynamic constraints
• Real-time approximation of optimal strategies for PDEs

Innovation

• Neural Operator predicts PDE system trajectories
• Proxy optimizer approximates optimal decision solutions
• Dual-network captures decision-PDE coupling in real-time