A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problems

TL;DR

This paper introduces a mesh-free, scalable deep learning approach for optimal control of obstacle problems, avoiding costly subproblem solutions and enabling efficient high-dimensional problem solving.

Contribution

It proposes a novel single-loop bilevel deep learning method with a new stochastic algorithm that removes the need for nested optimization and restrictive assumptions.

Findings

Achieves satisfactory accuracy in complex obstacle control problems.

Reduces computational cost compared to classical numerical methods.

Handles high-dimensional and irregular domain problems effectively.

Abstract

Optimal control of obstacle problems arises in a wide range of applications and is computationally challenging due to its nonsmoothness, nonlinearity, and bilevel structure. Classical numerical approaches rely on mesh-based discretization and typically require solving a sequence of costly subproblems. In this work, we propose a single-loop bilevel deep learning method, which is mesh-free, scalable to high-dimensional and complex domains, and avoids repeated solution of discretized subproblems. The method employs constraint-embedding neural networks to approximate the state and control and preserves the bilevel structure. To train the neural networks efficiently, we propose a Single-Loop Stochastic First-Order Bilevel Algorithm (S2-FOBA), which eliminates nested optimization and does not rely on restrictive lower-level uniqueness assumptions. We analyze the convergence behavior of S2-FOBA under mild assumptions. Numerical experiments on benchmark examples, including distributed and obstacle control problems with regular and irregular obstacles on complex domains, demonstrate that the proposed method achieves satisfactory accuracy while reducing computational cost compared to classical numerical methods.