Constructive Lyapunov Functions via Topology-Preserving Neural Networks

TL;DR

This paper introduces a neural network approach that constructs Lyapunov functions, achieving optimal convergence, efficiency, and scalability, with applications validated on large networks and improvements in transformer models.

Contribution

It presents a novel neural network method for constructing Lyapunov functions with proven guarantees and broad theoretical connections, transforming classical theorems into scalable algorithms.

Findings

99.75% improvement over baseline methods on semantic networks

Exponential convergence confirmed in large-scale networks

Enhanced transformer performance with 14.7% perplexity reduction

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.