Solving PDE-constrained Control Problems Using Operator Learning

TL;DR

This paper introduces a new framework that uses operator learning to efficiently solve PDE-constrained control problems by creating surrogate models, enabling rapid optimal control inference without heavy computations.

Contribution

It presents a two-phase approach combining operator learning and control optimization, applicable to both data-driven and data-free PDE control problems.

Findings

Successfully applied to diverse PDE control problems

Achieves faster control inference after training surrogate models

Works with various PDEs from Poisson to Burgers' equation

Abstract

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.