Constructive Lyapunov Functions via Topology-Preserving Neural Networks

• URL: http://arxiv.org/abs/2510.24730v1
• Date: Fri, 10 Oct 2025 17:46:52 GMT
• Title: Constructive Lyapunov Functions via Topology-Preserving Neural Networks
• Authors: Jaehong Oh,
• Abstract summary: ONN achieves order-optimal performance on convergence rate, edge efficiency, and computational complexity.<n> Empirical validation on 3M-node semantic networks demonstrates 99.75% improvement over baseline methods.<n>ORTSF integration into transformers achieves 14.7% perplexity reduction and 2.3 faster convergence.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove that ONN achieves order-optimal performance on convergence rate ($\mu \propto \lambda_2$), 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 ($\mu = 3.2 \times 10^{-4}$) 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.

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