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

This paper proposes a mesh-free, single-loop bilevel deep learning method using constraint-embedding neural networks and a novel S2-FOBA algorithm to efficiently solve optimal control of obstacle problems in high-dimensional and complex domains, achieving satisfactory accuracy while reducing computational costs compared to classical numerical approaches.

Original authors: Yongcun Song, Shangzhi Zeng, Jin Zhang, Lvgang Zhang

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yongcun Song, Shangzhi Zeng, Jin Zhang, Lvgang Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Two-Level Game

Imagine you are the CEO of a factory (the "Upper Level"). Your goal is to make a product that looks exactly like a specific blueprint you have in mind, while spending as little money as possible on raw materials.

However, you don't build the product yourself. You hire a master craftsman (the "Lower Level"). The craftsman has a very strict rule: they must build the product using a specific type of clay that has a hard, invisible floor underneath it. If the clay tries to go below this floor, it bounces back up. The craftsman will always try to build the shape that requires the least amount of energy to hold together, given the floor and the materials you provide.

The Problem:
You want to choose the materials (the "control") to get the best final product. But you can't just tell the craftsman, "Make it look like this." You have to trust that the craftsman will naturally build the most energy-efficient shape possible given your materials. If you try to force the shape, the craftsman will break the rules (the physics of the clay).

This is a Bilevel Problem:

Lower Level: The craftsman finds the best shape for a given set of materials.
Upper Level: You choose the materials to make that best shape look like your blueprint.

This is incredibly hard to solve because the "best shape" changes in a jerky, unpredictable way (it's "nonsmooth") when you change the materials slightly.

The Old Way: The Grid Trap

Traditionally, mathematicians solve this by drawing a giant grid (like a chessboard) over the factory floor. They calculate the clay's behavior on every single square of the grid.

The Flaw: If the floor is bumpy or the factory is shaped like a weird flower, the grid gets messy. You have to redraw the grid every time you change the materials. If the factory is huge (high-dimensional), the grid becomes so massive that computers crash trying to solve it. It's like trying to map the entire ocean by measuring every single drop of water with a ruler.

The New Way: The "Single-Loop" Deep Learning Method

The authors propose a new method using Deep Learning (AI) that avoids the grid entirely. Think of it as replacing the grid with a flexible, magical sheet that can stretch to fit any shape, no matter how complex.

Here is how their method works, broken down into three parts:

1. The "Constraint-Embedding" Neural Networks

Usually, AI models have to be told, "Don't go below the floor!" and they learn this through trial and error (penalties).

The Innovation: The authors built the "floor" directly into the AI's brain. They designed the neural networks (the AI models) so that it is physically impossible for them to suggest a shape that breaks the rules.
The Analogy: Instead of teaching a child not to touch a hot stove by yelling "Don't touch!", they put a physical barrier between the child and the stove. The child cannot touch it. This makes the training much faster and more reliable.

2. The "Single-Loop" Algorithm (S2-FOBA)

In the old way of solving these problems, the computer would have to play a game of "Guess and Check" repeatedly:

Guess a material.
Solve the craftsman's problem perfectly.
Check if the result is good.
If not, guess again.
This is like a nested loop (a game inside a game), which is slow and computationally expensive.

The authors created a new algorithm called S2-FOBA (Single-Loop Stochastic First-Order Bilevel Algorithm).

The Innovation: Instead of solving the craftsman's problem perfectly every single time, the algorithm takes a "good enough" step forward for both the CEO and the craftsman simultaneously.
The Analogy: Imagine two people walking up a mountain together. The old method would have Person A stop, wait for Person B to find the exact summit, then move. The new method has them both take small steps forward at the same time, constantly adjusting their path based on the terrain. They move in a single loop, making the process much faster and less likely to get stuck.

3. The Two-Stage Strategy

Because the "single-loop" method is so fast, it might leave the craftsman slightly off the perfect "energy-efficient" shape.

The Fix: The authors use a two-stage approach.
- Stage 1: Run the fast, single-loop training to get a very good starting point.
- Stage 2: Once the materials are chosen, run a quick, final check to ensure the craftsman's shape is perfectly optimized for those materials.
  This ensures the final result is both fast to find and physically accurate.

What Did They Prove?

The paper doesn't just say "it works"; they mathematically proved that this method will eventually converge to a solution (it won't run forever without finding an answer).

They tested their method on several scenarios:

Standard 2D Problems: Regular shapes and tricky "active" zones where the clay touches the floor.
High Dimensions: A 5-dimensional problem (which is impossible for the old grid methods). Their AI handled it easily.
Complex Shapes: A factory with a weird, flower-shaped boundary. The grid methods struggle here, but their "flexible sheet" AI handled it perfectly.

The Verdict

The authors claim their method is:

Mesh-free: No need for grids or rulers.
Scalable: Works in high dimensions where old methods fail.
Efficient: It doesn't get stuck in slow, nested loops.
Accurate: It respects the physics of the problem strictly.

In short, they replaced a slow, rigid, grid-based calculator with a fast, flexible, AI-driven guide that respects the rules of the game by design, not by force.

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

1. Problem Statement

The paper addresses the optimal control of obstacle problems, a class of nonsmooth, nonlinear, and hierarchically structured optimization problems. These problems arise in diverse physical and engineering applications, such as elastic-plastic torsion, membrane deformation, and American option pricing.

Mathematically, the problem is formulated as a bilevel optimization task:

Upper Level: Minimize an objective functional $J(y, u)$ (typically tracking a target state $y_d$ with control regularization) with respect to the control $u$ .
Lower Level: The state $y$ is not an independent variable but is implicitly defined as the unique minimizer of an energy functional $E(y', u)$ subject to an obstacle constraint $y' \geq \psi$ .

The core challenges include:

Nonsmoothness and Non-differentiability: The control-to-state operator is generally not Gâteaux-differentiable due to the obstacle constraint, complicating gradient-based optimization.
Free Boundary: The solution involves a free boundary (the interface between the contact and non-contact sets) which may not align with mesh grids in classical methods, leading to significant errors.
Computational Scalability: Classical numerical methods (e.g., Finite Element Methods, Finite Difference Methods) rely on mesh-based discretizations. Solving the resulting large-scale, ill-conditioned algebraic systems repeatedly is computationally expensive and struggles to scale to high-dimensional settings or complex geometries.
Limitations of Existing Deep Learning: Standard deep learning approaches for PDE control often collapse the bilevel structure into a single-level problem by weighting objectives. The authors argue this fails to enforce the strict hierarchical dependence of the state on the control, leading to physically inconsistent solutions.

2. Methodology

The proposed solution is a Single-Loop Bilevel Deep Learning Framework that avoids mesh discretization and preserves the problem's intrinsic structure.

2.1. Constraint-Embedding Neural Networks

Instead of using penalty terms or active-set identification to handle constraints, the authors design neural networks (NNs) that strictly enforce constraints by construction:

State Approximation ( $\hat{y}$ ): The network output is transformed (e.g., via squaring or ReLU operations combined with a smooth boundary function $m(x)$ ) to ensure $\hat{y} \geq \psi$ and $\hat{y} = 0$ on the boundary $\partial\Omega$ .
Control Approximation ( $\hat{u}$ ): Similarly, the control network is transformed to strictly satisfy box constraints $u_a \leq \hat{u} \leq u_b$ .

2.2. Stochastic Bilevel Formulation

The continuous problem is approximated by a finite-dimensional stochastic bilevel optimization problem over the NN parameters ( $\theta_y, \theta_u$ ). The integrals in the objective and energy functionals are approximated using Monte Carlo sampling (mini-batches) over the domain $\Omega$ , making the method mesh-free.

2.3. The S2-FOBA Algorithm

To solve the resulting stochastic bilevel problem, the authors develop the Single-Loop Stochastic First-Order Bilevel Algorithm (S2-FOBA). Key features include:

Moreau Envelope Reformulation: The lower-level constraint is reformulated using the Moreau envelope, which provides a smooth approximation of the value function. This avoids the need for restrictive assumptions like lower-level uniqueness or strong convexity.
Single-Loop Structure: Unlike traditional bilevel methods that require nested optimization (solving the lower-level problem to convergence at every upper-level step), S2-FOBA updates the lower-level parameters and the auxiliary variable tracking the proximal point simultaneously in a single loop.
Stochastic Updates: The algorithm uses stochastic gradients computed from mini-batches, avoiding the computational cost of solving large linear systems or computing exact gradients over the entire domain.
Two-Stage Training Strategy:
1. Bilevel Training: S2-FOBA is used to find approximate optimal parameters.
2. Lower-Level Refinement: The control parameters are fixed, and the lower-level problem is solved once more to ensure the state strictly satisfies the optimality condition, enhancing physical consistency.

3. Key Contributions

Framework Design: A bilevel deep learning framework that preserves the hierarchical structure of obstacle control problems, avoiding the pitfalls of weighted single-level formulations.
Constraint Embedding: Novel NN architectures that enforce obstacle and control constraints strictly without penalty tuning or active-set detection.
Algorithm Development: The S2-FOBA algorithm, which is:
- Single-loop: Eliminates nested optimization.
- Stochastic: Efficient for high dimensions via mini-batch sampling.
- Robust: Does not rely on lower-level uniqueness or strong convexity assumptions.
Convergence Analysis: Theoretical proof of convergence for S2-FOBA under mild assumptions (weak convexity of the lower-level objective), showing the descent of a merit function and the decay of a stationarity measure.
Extensibility: Demonstration that the method extends naturally to optimal control of general Elliptic Variational Inequalities (EVIs), including those with non-symmetric operators.

4. Numerical Results

The method was validated on several benchmark problems using Python and PyTorch on high-performance hardware (NVIDIA H100 GPU):

Example 1 (Bi-active set): A 2D problem with a non-trivial active set. The proposed method achieved significantly lower relative $L_2$ errors compared to a naive weighted single-level deep learning approach (Algorithm 1). It also demonstrated superior efficiency compared to the classical active-set method: while the active-set method's accuracy improved with mesh refinement (requiring re-computation), the proposed method maintained consistent error regardless of evaluation grid resolution and required only a forward pass for new resolutions.
Example 2 (Flat free boundary): A problem with a challenging flat transition in the active set. The method successfully captured the solution with robust accuracy at the free boundary, where primal active-set techniques often struggle.
Example 3 (High-dimensional): A 5-dimensional distributed optimal control problem. Using Separable Physics-Informed Neural Networks (SPINNs) and quasi-Monte Carlo sampling, the method successfully overcame the curse of dimensionality, accurately capturing optimal solutions.
Example 4 (Complex domain): A problem on a non-rectangular, flower-shaped domain. The mesh-free nature of the method allowed for seamless handling of the irregular geometry without complex mesh generation.
Extensions (Examples 5 & 6): The framework was successfully applied to obstacle control (where the obstacle itself is the control) and optimal control of non-symmetric EVIs, demonstrating broad applicability.

5. Significance and Claims

The paper claims that this work provides a computationally efficient and scalable learning-based method for optimal control problems that have been difficult to resolve with existing deep learning techniques.

Significance: The primary contribution is the ability to solve high-dimensional and complex-domain optimal control problems without the computational bottlenecks of mesh-based discretizations. By preserving the bilevel structure, the method ensures physical consistency that weighted single-level approaches lack.
Modesty: The authors acknowledge that while S2-FOBA is efficient, hyperparameter tuning (step sizes, penalty sequences) currently requires manual adjustment and could benefit from future automation. They also note that the framework is currently limited to stationary problems, with time-dependent obstacle problems and stochastic EVIs identified as directions for future work.
Comparison: The method is presented as outperforming existing deep learning approaches in terms of solution quality and being more efficient than classical numerical methods for high-dimensional or complex-domain scenarios, though classical methods may still offer higher precision for low-dimensional problems with fine meshes.