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

Efficient computation of the boundary of convex sets defined by mappings subject to explicit constraints is crucial for control, optimization, and multi-objective learning. However, existing methods often struggle to scale in high-dimensional spaces or under complex constraints. This paper introduces a learning-based framework, the Kolmogorov-Arnold Network enhanced Primal-Dual Neural Network (KAN-PDNN), which approximates the solution map of parameterized optimization problems that implicitly define such sets. By incorporating Karush-Kuhn-Tucker (KKT) conditions into a tailored loss function and employing an adaptive parameter sampling strategy, KAN-PDNN achieves accurate and comprehensive boundary reconstruction. We evaluate KAN-PDNN on high-dimensional convex and reachable set estimation tasks. Comparative experiments against state-of-the-art neural baselines and extensive ablation studies demonstrate its superior performance in both approximation accuracy and boundary coverage. These results establish KAN-PDNN as a scalable and generalizable framework for convex set computation in high-dimensional and constrained environments.