# Verifiable Error Bounds for Physics-Informed Neural KKL Observers

arXiv: https://arxiv.org/abs/2603.20434 · HTML: https://mlsanity.com/paper/2603.20434/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 26 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 8 of 8)

- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Towards gain tuning for numerical KKL observers](https://mlsanity.com/paper/2204.00318/) (2022)
- [Learning-based Design of Luenberger Observers for Autonomous Nonlinear Systems](https://mlsanity.com/paper/2210.01476/) (2022)
- [KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning](https://mlsanity.com/paper/2501.11655/) (2025)
- [On the existence of KKL observers with nonlinear contracting dynamics (Long Version)](https://mlsanity.com/paper/2402.16432/) (2024)

## Abstract (arXiv)

This paper proposes a computable state-estimation error bound for learning-based Kazantzis--Kravaris/Luenberger (KKL) observers. Recent work learns the KKL transformation map with a physics-informed neural network (PINN) and a corresponding left-inverse map with a conventional neural network. However, no computable state-estimation error bounds are currently available for this approach. We derive a state-estimation error bound that depends only on quantities that can be certified over a prescribed region using neural network verification. We further extend the result to bounded additive measurement noise and demonstrate the guarantees on nonlinear benchmark systems.