Solving PDE-constrained Control Problems Using Operator Learning
This work addresses the computational challenge of solving PDE-constrained control problems for applications in modeling complex physical systems, but it appears incremental as it builds on existing operator learning methods.
The authors tackled PDE-constrained optimal control problems by proposing a framework that uses surrogate models for PDE solution operators, enabling efficient inference of optimal control without intensive computations. They demonstrated successful applications to various problems, including the Poisson and Burgers' equations, though no concrete numerical results were provided.
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.