Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations
Boris Kramer, Gleb Pogudin
TL;DR
The chapter addresses turning nonlinear autonomous ODEs into polynomial or quadratic form by lifting with additional variables, enabling easier analysis, simulation, and learning. It presents existence theory for polynomialization via differential-algebraic function properties, and offers practical algorithms (BioCham and QBee) to compute lifted representations, illustrated on a sigmoid perceptron and a MAPK signaling model. The work highlights both theoretical results and practical tools, while identifying open problems such as optimality guarantees, extensions to DAEs/PDEs, and stability-preserving lifting. The contributions advance a unified view of structure discovery in dynamical systems and pave the way for improved model reduction, analytical tractability, and analog-computing interpretations through lifted polynomial dynamics.
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.