Deep mixed residual method for solving PDE-constrained optimization problems

The deep mixed residual method (DeepMRM) is a technique to solve partial differential equation. In this paper, it is applied to tackle PDE-constrained optimization problems (PDE-COPs). For a PDE-COP, we transform it into an optimality system, and then employ mixed residual method (MRM) on this system. By implementing the DeepMRM with three different network structures (fully connected neural network, residual network, and attention fully connected neural network), we successfully solve PDE-COPs including elliptic, semi-linear elliptic, and Navier-Stokes (NS) equation constrained optimization problems. Compared with the exact or high-fidelity solutions, the DeepMRM provides an effective approach for solving PDE-COPs using the three different network structures.