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

Yongcun Song, Shangzhi Zeng, Jin Zhang, Lvgang Zhang

TL;DR

This work tackles the challenging problem of optimal control for obstacle problems, which are nonsmooth and have a bilevel structure. It introduces constraint-embedding neural networks to preserve feasibility for state and control, and a single-loop stochastic bilevel training algorithm (S2-FOBA) built on a Moreau envelope reformulation to avoid nested optimization. A two-stage lower-level refinement ensures numerical feasibility of the state after training. The approach is mesh-free, scalable to high dimensions and complex domains, and yields competitive accuracy with reduced computational cost compared with classical discretization-based methods. The framework is extended to complex geometries and elliptic variational inequalities, highlighting its versatility and potential for broad applicability in PDE-constrained optimization.

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.

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

TL;DR

This work tackles the challenging problem of optimal control for obstacle problems, which are nonsmooth and have a bilevel structure. It introduces constraint-embedding neural networks to preserve feasibility for state and control, and a single-loop stochastic bilevel training algorithm (S2-FOBA) built on a Moreau envelope reformulation to avoid nested optimization. A two-stage lower-level refinement ensures numerical feasibility of the state after training. The approach is mesh-free, scalable to high dimensions and complex domains, and yields competitive accuracy with reduced computational cost compared with classical discretization-based methods. The framework is extended to complex geometries and elliptic variational inequalities, highlighting its versatility and potential for broad applicability in PDE-constrained optimization.

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.
Paper Structure (30 sections, 4 theorems, 60 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 30 sections, 4 theorems, 60 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model
  3. Classical Numerical Methods
  4. Deep Learning Methods
  5. Methodology
  6. Organization
  7. Preliminaries
  8. The Distributed Optimal Control of an Obstacle Problem
  9. Applications of Existing Deep Learning Approaches to Problem (\ref{['eq:distribution']})
  10. The Stationarity Condition-Based Approach
  11. Objective Combination Approach
  12. The Bilevel Deep Learning Method for Solving Problem (\ref{['eq:distribution']})
  13. Neural Network Approximations with Constraints Embedding
  14. The S2-FOBA for Problem (\ref{['eq: stochastic bi']})
  15. Moreau Envelope-Based Reformulation
  16. ...and 15 more sections

Key Result

Theorem 4.1

Suppose that $\gamma \in (0, \frac{1}{2\rho})$ and the step sizes are chosen as with $p \in ((q+1)/2, 1)$ and $q \in (1/2,1)$. Assume further that the penalty parameter $c_k$ is nondecreasing. If $\eta_0 \in (0, \frac{2}{L_{e} + 2/\gamma - \rho} )$, and $\alpha_0, \beta_0$ are sufficiently small, then the sequence of iterates $\{\theta^{k}\}$ generated by Algorithm alg:MEHA sat for some $C_\sigma

Figures (8)

  • Figure 1: Training trajectories of Algorithm \ref{['alg:Bilevel Deep Learning Method']} for Example 1
  • Figure 1: Numerical Results of Algorithm \ref{['alg:Bilevel Deep Learning Method']} for Example 3
  • Figure 2: Numerical results of Algorithm \ref{['alg:Bilevel Deep Learning Method']} for Example 1
  • Figure 2: Numerical results of Algorithm \ref{['alg:Bilevel Deep Learning Method']} for Example 4
  • Figure 3: Numerical results of Algorithm \ref{['alg:Bilevel Deep Learning Method']} for Example 1 with control constraints
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Proof 1
  • Proof 2: Proof of Theorem \ref{['thm:convergence']}
  • Remark 6.1