# Data-driven discovery of coordinates and governing equations

## TL;DR

This paper introduces a novel method combining autoencoders and sparse regression to simultaneously discover optimal coordinates and governing equations from high-dimensional data, enhancing interpretability and model simplicity.

## Contribution

It presents the first integrated approach to learn both coordinate transformations and sparse dynamical models simultaneously using deep learning and sparse regression.

## Key findings

- Successfully applied to high-dimensional systems with low-dimensional dynamics
- Achieves interpretable, parsimonious models
- Demonstrates improved discovery of governing equations

## Abstract

The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam's razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom autoencoder to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional dynamical systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. It is the first method of its kind to place the discovery of coordinates and models on an equal footing.