Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System
This work addresses elliptic optimal control problems, which are important in engineering and physics, but it is incremental as it applies neural networks to an existing optimality system framework.
The authors developed a neural network solver for elliptic optimal control problems with box constraints, using a coupled system from the first-order optimality conditions and deep neural networks to represent solutions. They provided L² error bounds for state, control, and adjoint variables based on network parameters and sampling points, and validated the method with numerical examples.
In this work, we investigate a neural network based solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order optimality system of the optimal control problem, and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme, and provide $L^2(Ω)$ error bounds on the state, control and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.