# Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2402.12360 · HTML: https://mlsanity.com/paper/2402.12360/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 73 references that are in the map.
- First posted on arXiv 19 Feb 2024.

## Builds on (most cited first, 17 of 17)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs](https://mlsanity.com/paper/2203.05337/) (2022)
- [Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data](https://mlsanity.com/paper/2205.13545/) (2022)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks](https://mlsanity.com/paper/2210.04338/) (2022)
- [Deep KKL: Data-driven Output Prediction for Non-Linear Systems](https://mlsanity.com/paper/2103.12443/) (2021)
- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
- [Learning Robust State Observers using Neural ODEs (longer version)](https://mlsanity.com/paper/2212.00866/) (2022)
- [Luenberger observers for discrete-time nonlinear systems](https://mlsanity.com/paper/2002.02674/) (2020)
- [Using learning to control artificial avatars in human motor coordination tasks](https://mlsanity.com/paper/1810.04191/) (2018)
- [Discrete-Time Nonlinear Feedback Linearization via Physics-Informed Machine Learning](https://mlsanity.com/paper/2303.08884/) (2023)
- [Functional observers with linear error dynamics for discrete-time nonlinear systems, with application to fault diagnosis](https://mlsanity.com/paper/2106.02185/) (2021)

## Built on by (most cited first, 8 of 8)

- [LoopTool: Closing the Data-Training Loop for Robust LLM Tool Calls](https://mlsanity.com/paper/2511.09148/) (2025)
- [RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators](https://mlsanity.com/paper/2406.05470/) (2024)
- [Behavior Injection: Preparing Language Models for Reinforcement Learning](https://mlsanity.com/paper/2505.18917/) (2025)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [PINN-Obs: Physics-Informed Neural Network-Based Observer for Nonlinear Dynamical Systems](https://mlsanity.com/paper/2507.06712/) (2025)
- [Designing Robust Software Sensors for Nonlinear Systems via Neural Networks and Adaptive Sliding Mode Control](https://mlsanity.com/paper/2507.06817/) (2025)
- [Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach](https://mlsanity.com/paper/2603.21568/) (2026)

## Abstract (arXiv)

We use Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation map by solving a system of inhomogeneous functional equations. The performance of the proposed PINN approach is assessed via two illustrative case studies for which the observer linearizing transformation map can be derived analytically. We also perform an uncertainty quantification analysis for the proposed PINN scheme and we compare it with conventional power-series numerical implementations, which rely on the computation of a power series solution.