Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations

2025-02-14

AI Summary

This work addresses the analytical, simulation, and control challenges posed by nonlinear ordinary differential equations (ODEs). We propose an automated quadratization method that transforms general non-polynomial systems into equivalent quadratic (or low-degree polynomial) forms via judicious introduction of auxiliary variables. The approach unifies algebraic transformation, symbolic computation, graph-theoretic modeling, and optimization-based search within a coherent framework; it systematically clarifies existence theory for quadratization and presents the first comparative evaluation of two major classes of automated algorithms. We innovatively validate the method on a single-layer neural network and a cellular signaling pathway model, successfully handling diverse dynamical systems from biology and machine learning. Results demonstrate substantial reductions in complexity for dynamical analysis, controller synthesis, and data-driven modeling—thereby enhancing both interpretability and learnability of the underlying systems.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Polynomial structure in ODEs
• Quadratic form simplification
• Automated discovery algorithms

Innovation

• Polynomialization of nonlinear systems
• Quadratization reduces system degree
• Automated discovery algorithms reviewed