Constructive Lyapunov Functions via Topology-Preserving Neural Networks
This provides a scalable algorithm for constructive stability analysis in neural networks, robotics, and distributed systems, though it builds on existing theoretical foundations.
The paper tackled the problem of constructing Lyapunov functions for stability analysis by developing a topology-preserving neural network (ONN) that achieves order-optimal convergence rates and efficiency, with empirical validation showing a 99.75% improvement over baselines on 3M-node networks and a 14.7% perplexity reduction in transformers.
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.