# KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning

arXiv: https://arxiv.org/abs/2501.11655 · HTML: https://mlsanity.com/paper/2501.11655/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 51 references that are in the map.
- First posted on arXiv 20 Jan 2025.

## Builds on (most cited first, 14 of 14)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [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)
- [Learning Robust State Observers using Neural ODEs (longer version)](https://mlsanity.com/paper/2212.00866/) (2022)
- [On the existence of KKL observers with nonlinear contracting dynamics (Long Version)](https://mlsanity.com/paper/2402.16432/) (2024)
- [Reconstructing Indistinguishable Solutions Via Set-Valued KKL Observer](https://mlsanity.com/paper/2301.09690/) (2023)
- [Joint vehicle state and parameters estimation via Twin-in-the-Loop observers](https://mlsanity.com/paper/2309.01461/) (2023)
- [Local Lipschitz Constant Computation of ReLU-FNNs: Upper Bound Computation with Exactness Verification](https://mlsanity.com/paper/2310.11104/) (2023)
- [Carleman Lifting for Nonlinear System Identification with Guaranteed Error Bounds](https://mlsanity.com/paper/2205.15009/) (2022)
- [Sensor Fault Detection and Isolation in Autonomous Nonlinear Systems Using Neural Network-Based Observers](https://mlsanity.com/paper/2304.08837/) (2023)

## Built on by (most cited first, 4 of 4)

- [Closed-Loop Neural Operator-Based Observer of Traffic Density](https://mlsanity.com/paper/2504.04873/) (2025)
- [HyperKKL: Enabling Non-Autonomous State Estimation through Dynamic Weight Conditioning](https://mlsanity.com/paper/2602.22630/) (2026)
- [Verifiable Error Bounds for Physics-Informed Neural KKL Observers](https://mlsanity.com/paper/2603.20434/) (2026)
- [HyperKKL: Learning KKL Observers for Non-Autonomous Nonlinear Systems via Hypernetwork-Based Input Conditioning](https://mlsanity.com/paper/2603.29744/) (2026)

## Abstract (arXiv)

This paper proposes a novel learning approach for designing Kazantzis-Kravaris or nonlinear Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher-dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose learning the forward mapping using a physics-informed neural network, and then learning its inverse mapping with a conventional feedforward neural network. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided, including non-asymptotic learning guarantees that link approximation quality to finite sample sizes. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples, showing better generalization capability outside the training domain compared to state-of-the-art methods.