# Deep KKL: Data-driven Output Prediction for Non-Linear Systems

arXiv: https://arxiv.org/abs/2103.12443 · HTML: https://mlsanity.com/paper/2103.12443/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 41 references that are in the map.
- First posted on arXiv 23 Mar 2021.

## Builds on (most cited first, 12 of 12)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge](https://mlsanity.com/paper/1711.07970/) (2017)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Neural Lander: Stable Drone Landing Control using Learned Dynamics](https://mlsanity.com/paper/1811.08027/) (2018)
- [A survey on intrinsic motivation in reinforcement learning](https://mlsanity.com/paper/1908.06976/) (2019)
- [CoPhy: Counterfactual Learning of Physical Dynamics](https://mlsanity.com/paper/1909.12000/) (2019)
- [EgoMap: Projective mapping and structured egocentric memory for Deep RL](https://mlsanity.com/paper/2002.02286/) (2020)
- [Visual Reaction: Learning to Play Catch with Your Drone](https://mlsanity.com/paper/1912.02155/) (2019)

## Built on by (most cited first, 10 of 10)

- [Filtered-CoPhy: Unsupervised Learning of Counterfactual Physics in Pixel Space](https://mlsanity.com/paper/2202.00368/) (2022)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)
- [Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.12360/) (2024)
- [Reasoning in visual navigation of end-to-end trained agents: a dynamical systems approach](https://mlsanity.com/paper/2503.08306/) (2025)
- [Learning Reduced Nonlinear State-Space Models: an Output-Error Based Canonical Approach](https://mlsanity.com/paper/2206.04791/) (2022)
- [Learning Hybrid Dynamics Models With Simulator-Informed Latent States](https://mlsanity.com/paper/2309.02873/) (2023)
- [HyperKKL: Enabling Non-Autonomous State Estimation through Dynamic Weight Conditioning](https://mlsanity.com/paper/2602.22630/) (2026)
- [A mathematical framework for time-delay reservoir computing analysis](https://mlsanity.com/paper/2603.18706/) (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)

We address the problem of output prediction, ie. designing a model for autonomous nonlinear systems capable of forecasting their future observations. We first define a general framework bringing together the necessary properties for the development of such an output predictor. In particular, we look at this problem from two different viewpoints, control theory and data-driven techniques (machine learning), and try to formulate it in a consistent way, reducing the gap between the two fields. Building on this formulation and problem definition, we propose a predictor structure based on the Kazantzis-Kravaris/Luenberger (KKL) observer and we show that KKL fits well into our general framework. Finally, we propose a constructive solution for this predictor that solely relies on a small set of trajectories measured from the system. Our experiments show that our solution allows to obtain an efficient predictor over a subset of the observation space.