Title: Optimizing Obstacle Problems via a Single-Loop Bilevel Deep Learning Framework

Abstract:

The optimal control of obstacle problems presents significant computational hurdles stemming from their inherent nonsmoothness, nonlinearity, and bilevel architecture, despite their prevalence across numerous applications. Traditional numerical techniques typically depend on mesh-based discretization, necessitating the resolution of a series of expensive subproblems. To address these limitations, this study introduces a novel single-loop bilevel deep learning approach. This method is mesh-free, capable of scaling to high-dimensional and intricate domains, and circumvents the need to repeatedly solve discretized subproblems. By utilizing constraint-embedding neural networks to estimate both the state and control variables, the proposed framework maintains the original bilevel structure. For efficient network training, we develop the Single-Loop Stochastic First-Order Bilevel Algorithm (S2-FOBA). This algorithm removes the requirement for nested optimization and operates without relying on restrictive assumptions regarding the uniqueness of lower-level solutions. We provide a convergence analysis for S2-FOBA under mild conditions. Benchmark tests, encompassing distributed and obstacle control scenarios involving both regular and irregular obstacles on complex geometries, indicate that our method delivers competitive accuracy while significantly lowering computational expenses relative to classical numerical strategies.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC