| Abstract: |
| We present neural network methods for solving non-smooth PDE-constrained optimization problems. Our investigation focuses on three challenging categories: (1) optimization with non-smooth regularization, (2) optimal control of PDEs involving interfaces, and (3) optimal control of elliptic variational inequalities. For each category, we develop tailored neural network algorithms that exploit the specific mathematical structure of the underlying problem. The principal advantages of our methods are that they are mesh-free, thus avoiding grid generation challenges; computationally scalable to high dimensions and complex domains; and straightforward to implement. Extensive numerical experiments demonstrate their computational efficiency and accuracy on benchmark problems |
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