{"ID":2857697,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.24730","arxiv_id":"2510.24730","title":"Constructive Lyapunov Functions via Topology-Preserving Neural Networks","abstract":"We prove that ONN achieves order-optimal performance on convergence rate ($μ\\propto λ_2$), edge efficiency ($E = N$ for minimal connectivity $k = 2$), and computational complexity ($O(N d^2)$). Empirical validation on 3M-node semantic networks demonstrates 99.75\\% improvement over baseline methods, confirming exponential convergence ($μ= 3.2 \\times 10^{-4}$) and topology preservation. ORTSF integration into transformers achieves 14.7\\% perplexity reduction and 2.3 faster convergence on WikiText-103. We establish deep connections to optimal control (Hamilton-Jacobi-Bellman), information geometry (Fisher-efficient natural gradient), topological data analysis (persistent homology computation in $O(KN)$), discrete geometry (Ricci flow), and category theory (adjoint functors). This work transforms Massera's abstract existence theorem into a concrete, scalable algorithm with provable guarantees, opening pathways for constructive stability analysis in neural networks, robotics, and distributed systems.","short_abstract":"We prove that ONN achieves order-optimal performance on convergence rate ($μ\\propto λ_2$), edge efficiency ($E = N$ for minimal connectivity $k = 2$), and computational complexity ($O(N d^2)$). Empirical validation on 3M-node semantic networks demonstrates 99.75\\% improvement over baseline methods, confirming exponenti...","url_abs":"https://arxiv.org/abs/2510.24730","url_pdf":"https://arxiv.org/pdf/2510.24730v1","authors":"[\"Jaehong Oh\"]","published":"2025-10-10T17:46:52Z","proceeding":"eess.SY","tasks":"[\"eess.SY\",\"cs.LG\"]","methods":"[\"Transformer\",\"Large Language Model\"]","has_code":false}