# Knowledge-Based Learning of Nonlinear Dynamics and Chaos

arXiv: https://arxiv.org/abs/2010.03415 · HTML: https://mlsanity.com/paper/2010.03415/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2020.

## Built on by (most cited first, 14 of 14)

- [Investigating Compounding Prediction Errors in Learned Dynamics Models](https://mlsanity.com/paper/2203.09637/) (2022)
- [NODEO: A Neural Ordinary Differential Equation Based Optimization Framework for Deformable Image Registration](https://mlsanity.com/paper/2108.03443/) (2021)
- [Learning continuous models for continuous physics](https://mlsanity.com/paper/2202.08494/) (2022)
- [KNODE-MPC: A Knowledge-based Data-driven Predictive Control Framework for Aerial Robots](https://mlsanity.com/paper/2109.04821/) (2021)
- [Online Dynamics Learning for Predictive Control with an Application to Aerial Robots](https://mlsanity.com/paper/2207.09344/) (2022)
- [Learning Globally Smooth Functions on Manifolds](https://mlsanity.com/paper/2210.00301/) (2022)
- [Flying Quadrotors in Tight Formations using Learning-based Model Predictive Control](https://mlsanity.com/paper/2410.09727/) (2024)
- [Knowledge-based Neural Ordinary Differential Equations for Cosserat Rod-based Soft Robots](https://mlsanity.com/paper/2408.07776/) (2024)
- [Online Adaptation for Flying Quadrotors in Tight Formations](https://mlsanity.com/paper/2506.17488/) (2025)
- [LEARNEST: LEARNing Enhanced Model-based State ESTimation for Robots using Knowledge-based Neural Ordinary Differential Equations](https://mlsanity.com/paper/2209.08185/) (2022)
- [Learning-enhanced Nonlinear Model Predictive Control using Knowledge-based Neural Ordinary Differential Equations and Deep Ensembles](https://mlsanity.com/paper/2211.13829/) (2022)
- [Leveraging Predictive Models for Adaptive Sampling of Spatiotemporal Fluid Processes](https://mlsanity.com/paper/2304.00732/) (2023)
- [External-Wrench Estimation for Aerial Robots Exploiting a Learned Model](https://mlsanity.com/paper/2504.08156/) (2025)
- [Conditioning on PDE Parameters to Generalise Deep Learning Emulation of Stochastic and Chaotic Dynamics](https://mlsanity.com/paper/2509.09599/) (2025)

## Abstract (arXiv)

Extracting predictive models from nonlinear systems is a central task in scientific machine learning. One key problem is the reconciliation between modern data-driven approaches and first principles. Despite rapid advances in machine learning techniques, embedding domain knowledge into data-driven models remains a challenge. In this work, we present a universal learning framework for extracting predictive models from nonlinear systems based on observations. Our framework can readily incorporate first principle knowledge because it naturally models nonlinear systems as continuous-time systems. This both improves the extracted models' extrapolation power and reduces the amount of data needed for training. In addition, our framework has the advantages of robustness to observational noise and applicability to irregularly sampled data. We demonstrate the effectiveness of our scheme by learning predictive models for a wide variety of systems including a stiff Van der Pol oscillator, the Lorenz system, and the Kuramoto-Sivashinsky equation. For the Lorenz system, different types of domain knowledge are incorporated to demonstrate the strength of knowledge embedding in data-driven system identification.