Constructive Lyapunov Functions via Topology-Preserving Neural Networks

TL;DR

This work reframes the classical inverse-Lyapunov problem by introducing ontological topology-preserving neural networks (ONN) that replace trajectory-based Lyapunov constructions with topology-derived invariants. The total ONN loss combines consensus, curvature, and topology terms to form a topologically constructive Lyapunov function with explicit class-$\mathcal{K}_\infty$ bounds, enabling exponential convergence at a rate tied to the graph’s spectral gap $\lambda_2(\mathcal{L})$. It extends stability theory to non-smooth, hybrid dynamics via Fejér-monotone arguments under discrete surgery, and provides a global ROA characterization through persistent homology, yielding a computable boundary defined by Betti numbers. The framework also delivers explicit delay margins (ORTSF) and ISS guarantees for delayed ONN, with strong empirical validation at 3M-node scale and transformer integration showing practical improvements. Collectively, the paper bridges constructive Lyapunov theory, topology, and modern neural networks to deliver scalable stability guarantees with real-world applicability in large-scale, topology-rich systems.

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.

Figures (14)

• Figure 1: Theorem dependency graph for constructive Lyapunov theory via ONN. Arrows indicate logical dependencies: Theorem \ref{['thm:onn_topologically_constructive']} (this result) synthesizes classical foundations (Razumikhin, Massera, Kurzweil) with ONN-specific extensions (Class-$\mathcal{K}_\infty$ bounds, surgery, delay robustness). The graph reveals how non-constructive existence theorems (Sec. II--III) are transformed into computable, polynomial-time algorithms (Sec. IV).
• Figure 2: ORTSF delay margin derivation and validation. (a) Maximum tolerable delay $\tau_{\max}$ as a function of spectral gap $\mu$ for different Lipschitz constants $L$. The red star indicates the empirical configuration from Section \ref{['sec:3m_validation']} ($\mu = 3.2 \times 10^{-4}$, $L = 5$, $\tau_{\max} = 177$$\mu$s). (b) Convergence rate degradation: the delay-degraded rate $\tilde{\mu}$ decreases linearly with delay until instability at $\tau = \tau_{\max}$. Higher $\mu/L$ ratios provide better delay tolerance. (c) Proof sketch showing Razumikhin condition: delayed Lyapunov function $V(t - \tau)$ must satisfy $V(t - s) \leq q V(t)$ for all $s \in [0, \tau]$ to guarantee stability. The dimensional analysis confirms $[\tau_{\max}] = \text{seconds}$, consistent with physical time units.
• Figure 3: Comprehensive experimental setup for 3M-node semantic network validation. (a) System architecture: 512 NVIDIA A100 GPUs with 40 TB/s NVLink interconnect execute ONN dynamics (semantic flow + topology surgery) on a 3M-node network, evaluating convergence, spectral gap, and topology preservation. (b) Quantitative configuration: Complete specification of hardware, dataset parameters, ONN hyperparameters, and derived quantities. The empirical convergence rate $\mu_{\text{emp}} = 3.2 \times 10^{-4}$ is three orders of magnitude faster than the theoretical worst-case bound $\mu_{\text{theory}} = 2.5 \times 10^{-7}$, validating that theory provides conservative guarantees. (c) Computational breakdown: 47 seconds per iteration, dominated by semantic flow gradient computation (28s, 59.6%) and topology surgery (15s, 31.9%). (d) Validation metrics timeline: Exponential loss decay confirms $\mu = 3.2 \times 10^{-4}$, spectral gap stabilizes at $\lambda_2 \approx 10^{-6}$, and topology invariants ($\beta_0=1$, $\beta_1=999$) remain constant despite 60% surgery rate. This provides the quantitative basis for all empirical claims in Section \ref{['sec:empirical_validation']}.
• Figure 4: Topology stability for 3M-node ONN. Betti numbers $\beta_0$ (connectivity) and $\beta_1$ (genus) remain constant despite 60% surgery rate. Shaded regions show $\pm 1$ standard deviation over 5 trials.
• Figure 5: 3M-node validation dashboard. Top-left: Exponential convergence of total loss ($\mu = 3.2 \times 10^{-4}$). Top-right: Spectral gap $\lambda_2$ evolution, stabilizing at $\lambda_2 \approx 10^{-6}$. Bottom-left: Surgery rate (60%) and edge change distribution. Bottom-right: Computational performance (512 A100 GPUs, 47 seconds per iteration).

Theorems & Definitions (185)

• Theorem 1.2: Informal Statement of Theorem \ref{['thm:surgery_fejer_revised']}
• Theorem 1.3: Informal Statement of Theorem \ref{['thm:topological_roa_characterization']}
• Theorem 1.4: Informal Statement of Theorem \ref{['thm:ortsf_delay_margin']}
• Definition 2.1: Stability
• Definition 2.2: Asymptotic Stability
• Definition 2.3: Exponential Stability
• Definition 2.4: Topology-Preserving Dynamical Systems
• Remark 2.5: ODE to Graph Embedding Justification
• Definition 2.6: Lyapunov Function
• Theorem 2.7: Lyapunov's Direct Method