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

arXiv: https://arxiv.org/abs/2601.04120 · HTML: https://mlsanity.com/paper/2601.04120/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 59 references that are in the map.
- First posted on arXiv 7 Jan 2026.

## Builds on (most cited first, 19 of 19)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Deep Neural Network Structures Solving Variational Inequalities](https://mlsanity.com/paper/1808.07526/) (2018)
- [Fast PDE-constrained optimization via self-supervised operator learning](https://mlsanity.com/paper/2110.13297/) (2021)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators](https://mlsanity.com/paper/2305.20053/) (2023)
- [Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients](https://mlsanity.com/paper/2209.07075/) (2022)
- [Moreau Envelope for Nonconvex Bi-Level Optimization: A Single-loop and Hessian-free Solution Strategy](https://mlsanity.com/paper/2405.09927/) (2024)
- [Moreau Envelope Based Difference-of-weakly-Convex Reformulation and Algorithm for Bilevel Programs](https://mlsanity.com/paper/2306.16761/) (2023)
- [A physics-informed neural network framework for modeling obstacle-related equations](https://mlsanity.com/paper/2304.03552/) (2023)
- [Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems](https://mlsanity.com/paper/2307.00296/) (2023)
- [The Hard-Constraint PINNs for Interface Optimal Control Problems](https://mlsanity.com/paper/2308.06709/) (2023)
- [A neural network approach to learning solutions of a class of elliptic variational inequalities](https://mlsanity.com/paper/2411.18565/) (2024)
- [Prox-PINNs: A Deep Learning Algorithmic Framework for Elliptic Variational Inequalities](https://mlsanity.com/paper/2505.14430/) (2025)
- [An Operator Learning Approach to Nonsmooth Optimal Control of Nonlinear PDEs](https://mlsanity.com/paper/2409.14417/) (2024)
- [On the optimal control of some nonsmooth distributed parameter systems arising in mechanics](https://mlsanity.com/paper/1711.08418/) (2017)

## Abstract (arXiv)

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.