Constructive Lyapunov Functions via Topology-Preserving Neural Networks

Abstract

We prove that ONN achieves order-optimal performance on convergence rate ($\hat{I}¼\propto \hat{I}{»}_2$), 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 ($\hat{I}¼=3.2\times 10^{-4}$) 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.

54pages, 14 figures