# Learning Robust State Observers using Neural ODEs (longer version)

arXiv: https://arxiv.org/abs/2212.00866 · HTML: https://mlsanity.com/paper/2212.00866/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 31 references that are in the map.
- First posted on arXiv 1 Dec 2022.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory](https://mlsanity.com/paper/2007.06007/) (2020)
- [Towards gain tuning for numerical KKL observers](https://mlsanity.com/paper/2204.00318/) (2022)
- [Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs](https://mlsanity.com/paper/2201.05715/) (2022)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Learning-based Design of Luenberger Observers for Autonomous Nonlinear Systems](https://mlsanity.com/paper/2210.01476/) (2022)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)

## Built on by (most cited first, 7 of 7)

- [Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.12360/) (2024)
- [KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning](https://mlsanity.com/paper/2501.11655/) (2025)
- [Stable and Safe Human-aligned Reinforcement Learning through Neural Ordinary Differential Equations](https://mlsanity.com/paper/2401.13148/) (2024)
- [Designing Robust Software Sensors for Nonlinear Systems via Neural Networks and Adaptive Sliding Mode Control](https://mlsanity.com/paper/2507.06817/) (2025)
- [HyperKKL: Enabling Non-Autonomous State Estimation through Dynamic Weight Conditioning](https://mlsanity.com/paper/2602.22630/) (2026)
- [HyperKKL: Learning KKL Observers for Non-Autonomous Nonlinear Systems via Hypernetwork-Based Input Conditioning](https://mlsanity.com/paper/2603.29744/) (2026)
- [A KKL Observer Perspective on Reservoir Computing](https://mlsanity.com/paper/2610.04343/) (2026)

## Abstract (arXiv)

Relying on recent research results on Neural ODEs, this paper presents a methodology for the design of state observers for nonlinear systems based on Neural ODEs, learning Luenberger-like observers and their nonlinear extension (Kazantzis-Kravaris-Luenberger (KKL) observers) for systems with partially-known nonlinear dynamics and fully unknown nonlinear dynamics, respectively. In particular, for tuneable KKL observers, the relationship between the design of the observer and its trade-off between convergence speed and robustness is analysed and used as a basis for improving the robustness of the learning-based observer in training. We illustrate the advantages of this approach in numerical simulations.