# Deep Learning of Conjugate Mappings

arXiv: https://arxiv.org/abs/2104.01874 · HTML: https://mlsanity.com/paper/2104.01874/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 86 references that are in the map.
- First posted on arXiv 1 Apr 2021.

## Builds on (most cited first, 23 of 23)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Learning Symbolic Physics with Graph Networks](https://mlsanity.com/paper/1909.05862/) (2019)
- [SINDy-BVP: Sparse Identification of Nonlinear Dynamics for Boundary Value Problems](https://mlsanity.com/paper/2005.10756/) (2020)
- [Discovering conservation laws from data for control](https://mlsanity.com/paper/1811.00961/) (2018)
- [Sparse Identification of Slow Timescale Dynamics](https://mlsanity.com/paper/2006.00940/) (2020)
- [Data-Driven Stabilization of Periodic Orbits](https://mlsanity.com/paper/2010.13896/) (2020)

## Built on by (most cited first, 5 of 5)

- [Deep Learning Enhanced Dynamic Mode Decomposition](https://mlsanity.com/paper/2108.04433/) (2021)
- [Phase2vec: Dynamical systems embedding with a physics-informed convolutional network](https://mlsanity.com/paper/2212.03857/) (2022)
- [Manifold embedding data-driven mechanics](https://mlsanity.com/paper/2112.09842/) (2021)
- [Let's do the time-warp-attend: Learning topological invariants of dynamical systems](https://mlsanity.com/paper/2312.09234/) (2023)
- [Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences](https://mlsanity.com/paper/2503.10336/) (2025)

## Abstract (arXiv)

Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincar'e first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincar'e map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincar'e mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the R"ossler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto--Sivashinsky equation.