mlsanityData to 7 Oct 2026

Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification

arXiv 2312.09131 ↗First posted 14 Dec 2023math.OC cs.LG cs.SYcited by 20

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Abstract (arXiv)

We provide a systematic investigation of using physics-informed neural networks to compute Lyapunov functions. We encode Lyapunov conditions as a partial differential equation (PDE) and use this for training neural network Lyapunov functions. We analyze the analytical properties of the solutions to the Lyapunov and Zubov PDEs. In particular, we show that employing the Zubov equation in training neural Lyapunov functions can lead to approximate regions of attraction close to the true domain of attraction. We also examine approximation errors and the convergence of neural approximations to the unique solution of Zubov's equation. We then provide sufficient conditions for the learned neural Lyapunov functions that can be readily verified by satisfiability modulo theories (SMT) solvers, enabling formal verification of both local stability analysis and region-of-attraction estimates in the large. Through a number of nonlinear examples, ranging from low to high dimensions, we demonstrate that the proposed framework can outperform traditional sums-of-squares (SOS) Lyapunov functions obtained using semidefinite programming (SDP).

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 15 of its 61 references that are in the map. First posted on arXiv 14 Dec 2023.

Builds on

15 of 61 references.

References · 15

  1. 2022Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methodscited by 148
  2. 2020Neural Lyapunov Controlcited by 142
  3. 2021Lyapunov-stable neural-network controlcited by 66
  4. 2022Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guaranteescited by 39
  5. 2020Learning Lyapunov Functions for Hybrid Systemscited by 24
  6. 2021Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximationcited by 16
  7. 2021Learning Region of Attraction for Nonlinear Systemscited by 16
  8. 2020Computing Lyapunov functions using deep neural networkscited by 14
  9. 2020Automated and Sound Synthesis of Lyapunov Functions with SMT Solverscited by 10
  10. 2021Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equationcited by 7
  11. 2023Towards Learning and Verifying Maximal Neural Lyapunov Functionscited by 7
  12. 2020Overcoming the curse of dimensionality for approximating Lyapunov functions with deep neural networks under a small-gain conditioncited by 4
3 more
  1. 2020Smooth Converse Lyapunov-Barrier Theorems for Asymptotic Stability with Safety Constraints and Reach-Avoid-Stay Specificationscited by 4
  2. 2021Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systemscited by 3
  3. 2015A Sum-of-Squares approach to the Stability and Control of Interconnected Systems using Vector Lyapunov Functionscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 8; 2025: 7; 2026: 5.

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  1. 2024LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attractioncited by 10
  2. 2024Actor-Critic Physics-informed Neural Lyapunov Controlcited by 6
  3. 2025Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansioncited by 6
  4. 2024Deep Learning for Computing Convergence Rates of Markov Chainscited by 4
  5. 2024Formally Verified Physics-Informed Neural Control Lyapunov Functionscited by 4
  6. 2024Certified Training with Branch-and-Bound for Lyapunov-stable Neural Controlcited by 3
  7. 2025Safe Domains of Attraction for Discrete-Time Nonlinear Systems: Characterization and Verifiable Neural Network Estimationcited by 3
  8. 2025Deep Learning for Markov Chains: Lyapunov Functions, Poisson's Equation, and Stationary Distributionscited by 2
  9. 2025Off Policy Lyapunov Stability in Reinforcement Learningcited by 2
  10. 2024Manifold-Guided Lyapunov Control with Diffusion Modelscited by 1
  11. 2024Learning Koopman-based Stability Certificates for Unknown Nonlinear Systemscited by 1
  12. 2026Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorialcited by 1
8 more
  1. 2024(Un)supervised Learning of Maximal Lyapunov Functionscited by 0
  2. 2025Neural Control Barrier Functions from Physics Informed Neural Networkscited by 0
  3. 2025Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustnesscited by 0
  4. 2025Physics-Informed Machine Learning for Characterizing System Stabilitycited by 0
  5. 2026Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimationcited by 0
  6. 2026Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guaranteescited by 0
  7. 2026Verifiable Error Bounds for Physics-Informed Neural Network Solutions of Lyapunov and Hamilton-Jacobi-Bellman Equationscited by 0
  8. 2026Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flowscited by 0