Solving Elliptic Optimal Control Problems using Physics Informed Neural Networks

Optimal control problems subject to partial differential equations (PDE) constraints represent a very important class of problems in practice and have found various important applications in science and engineering, e.g., fluid flow, heat conduction, structural optimization, microelectronics, crystal growth, vascular surgery, and cardiac medicine, to name just a few. The mathematical theory of distributed / boundary optimal control problems subject to linear / nonlinear elliptic / parabolic PDEs is well understood (see, e.g., the monographs [27, 39, 30]). Numerically, one of the most common approaches to solve such problems utilizes the first-order optimality system and introduces either adjoint state or Lagrangian multiplier, and then iteratively updates the control using established optimization algorithms, e.g., (accelerated / stochastic) gradient descent method, semismooth Newton method, and alternating direction method of multipliers. In practical implementation, the resulting continuous formulation is then often discretized by finite element method (FEM) and spectral methods. In the past few years, deep neural networks (DNNs) have demonstrated remarkable performance across a wide range of applications, e.g., computer vision, imaging, and natural language processing. These successes have permeated into several scientific disciplines, and DNNs have also become very popular tools to approximate solutions to various PDEs. The ease of implementation, and the potential to overcome the notorious curse of dimensionality have sparked intensive interest in revisiting classical computational problems from a deep-learning perspective. In the last few years, a large number of neural solvers based on DNNs have been proposed for solving PDEs, e.g., physics informed neural network (PINN) [34], deep Ritz method [14], deep Galerkin method [36], weak adversarial networks [41], and deep least-squares method [9].