Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations

TL;DR

This paper reviews methods for transforming complex nonlinear dynamical systems into polynomial or quadratic forms, facilitating analysis and control, and discusses algorithms and examples for automating this process.

Contribution

It summarizes the current state of the art in discovering polynomial and quadratic structures in dynamical systems, including algorithms and practical examples.

Findings

Polynomialization reveals new system structures.

Two main algorithms automate the discovery process.

Examples include neural networks and cell signaling models.

Abstract

Dynamical systems with quadratic or polynomial drift exhibit complex dynamics, yet compared to nonlinear systems in general form, are often easier to analyze, simulate, control, and learn. Results going back over a century have shown that the majority of nonpolynomial nonlinear systems can be recast in polynomial form, and their degree can be reduced further to quadratic. This process of polynomialization/quadratization reveals new variables (in most cases, additional variables have to be added to achieve this) in which the system dynamics adhere to that specific form, which leads us to discover new structures of a model. This chapter summarizes the state of the art for the discovery of polynomial and quadratic representations of finite-dimensional dynamical systems. We review known existence results, discuss the two prevalent algorithms for automating the discovery process, and give examples in form of a single-layer neural network and a phenomenological model of cell signaling.