Learning convex set boundaries via primal-dual neural approximation with application to reachable set computation.

Area of Science:

• Computational mathematics
• Machine learning
• Optimization theory

Background:

• Efficient computation of convex set boundaries is vital for control, optimization, and multi-objective learning.
• Existing methods face challenges with scalability in high-dimensional spaces and complex constraints.

Purpose of the Study:

• To introduce a novel learning-based framework, KAN-PDNN, for approximating solution maps of parameterized optimization problems.
• To accurately and comprehensively reconstruct boundaries of convex sets under explicit constraints.

Main Methods:

• Developed a Kolmogorov-Arnold Network enhanced Primal-Dual Neural Network (KAN-PDNN).
• Integrated Karush-Kuhn-Tucker (KKT) conditions into a specialized loss function.
• Employed an adaptive parameter sampling strategy for enhanced boundary reconstruction.

Main Results:

• KAN-PDNN demonstrated superior performance in approximating convex and reachable set boundaries.
• Achieved high accuracy and comprehensive boundary coverage in high-dimensional tasks.
• Outperformed state-of-the-art neural baselines in comparative experiments.

Conclusions:

• KAN-PDNN provides a scalable and generalizable framework for convex set computation.
• The method is effective in high-dimensional and constrained environments.
• Advances the field of efficient boundary computation for complex mathematical sets.