# Markov Neural Operators for Learning Chaotic Systems

**Authors:** Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar  
**arXiv:** [2106.06898](https://arxiv.org/abs/2106.06898)  
**Date:** 2021-06-13  
**Summary page:** https://www.summarizepaper.com/en/paper/2106.06898v1/

## Key points

- Chaotic systems are difficult to predict due to their inherent instability and accumulation of small errors.
- Trajectories of chaotic systems can reside within a low-dimensional subspace called an attractor.
- Markovian chaotic systems can have their attractor uniquely determined by the Markov operator.
- Learning the Markov operator enables forecasting the overall behavior of chaotic systems.
- Neural operators, specifically Markov neural operators (MNO), can be used for learning chaotic systems.
- MNOs are trained using local one-step evolution information and can derive the global attractor and invariant measure of the system.
- The Markov neural operator formed a discrete semigroup that remained stable during empirical experiments.
- Neural operators are more accurate and stable compared to previous methods for predicting chaotic systems like Kuramoto-Sivashinsky and Navier-Stokes equations.
- This breakthrough has applications in various fields including physics, biology, and climate science.
- Learning the Markov operator through neural operators provides valuable insights into chaotic systems and allows predictions even without precise trajectory information.

## Summary

In the field of chaos theory, predicting the behavior of chaotic systems has always been a daunting task due to their inherent instability. The accumulation of small errors in simulating each time step often leads to completely different trajectories, making accurate predictions nearly impossible. However, researchers have discovered that the trajectories of many well-known chaotic systems actually reside within a low-dimensional subspace called an attractor. Remarkably, if a chaotic system is Markovian, meaning its future state depends solely on its current state and not on its past states, the attractor can be uniquely determined by a mathematical entity known as the Markov operator. This operator maps the evolution of infinitesimal time steps and provides valuable insights into the system's dynamics. Even though we may not be able to predict the exact trajectory of a chaotic system, learning this Markov operator enables us to forecast its overall behavior. Recently, a groundbreaking framework called neural operators has emerged for learning resolution-invariant solution operators for partial differential equations (PDEs). Building upon this framework, a team of researchers introduced a novel approach called Markov neural operators (MNO) specifically designed for learning chaotic systems. In their work, they trained an MNO using only local one-step evolution information. By composing the learned operator obtained from MNO, they were able to derive the global attractor and invariant measure of the chaotic system under study. Importantly, they observed that the Markov neural operator formed a discrete semigroup that neither collapsed nor blew up during their empirical experiments. The results demonstrated that neural operators are more accurate and stable compared to previous methods when applied to chaotic systems such as the Kuramoto-Sivashinsky and Navier-Stokes equations. This breakthrough opens up new possibilities for predicting and understanding complex dynamical phenomena in various fields including physics, biology, and climate science. Overall, this research highlights how learning the Markov operator through neural operators can provide valuable insights into chaotic systems, enabling us to make predictions about their behavior even in the absence of precise trajectory information. The potential applications of this approach are vast and could revolutionize our understanding of complex systems.