# Verifiable Error Bounds for Physics-Informed Neural Network Solutions of Lyapunov and Hamilton-Jacobi-Bellman Equations

arXiv: https://arxiv.org/abs/2603.19545 · HTML: https://mlsanity.com/paper/2603.19545/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 27 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation](https://mlsanity.com/paper/2109.13359/) (2021)
- [Computing Lyapunov functions using deep neural networks](https://mlsanity.com/paper/2005.08965/) (2020)
- [Physics-Informed Neural Network Policy Iteration: Algorithms, Convergence, and Verification](https://mlsanity.com/paper/2402.10119/) (2024)
- [LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attraction](https://mlsanity.com/paper/2403.10013/) (2024)
- [Towards Learning and Verifying Maximal Neural Lyapunov Functions](https://mlsanity.com/paper/2304.07215/) (2023)
- [Formally Verified Physics-Informed Neural Control Lyapunov Functions](https://mlsanity.com/paper/2409.20528/) (2024)
- [A Physics-Informed Learning Framework to Solve the Infinite-Horizon Optimal Control Problem](https://mlsanity.com/paper/2505.21842/) (2025)

## Abstract (arXiv)

Many core problems in nonlinear systems analysis and control can be recast as solving partial differential equations (PDEs) such as Lyapunov and Hamilton-Jacobi-Bellman (HJB) equations. Physics-informed neural networks (PINNs) have emerged as a promising mesh-free approach for approximating their solutions, but in most existing works there is no rigorous guarantee that a small PDE residual implies a small solution error. This paper develops verifiable error bounds for approximate solutions of Lyapunov and HJB equations, with particular emphasis on PINN-based approximations. For both the Lyapunov and HJB PDEs, we show that a verifiable residual bound yields relative error bounds with respect to the true solutions as well as computable a posteriori estimates in terms of the approximate solutions. For the HJB equation, this also yields certified upper and lower bounds on the optimal value function on compact sublevel sets and quantifies the optimality gap of the induced feedback policy. We further show that one-sided residual bounds already imply that the approximation itself defines a valid Lyapunov or control Lyapunov function. We illustrate the results with numerical examples.