Data-driven discovery of polynomial ODEs with provably bounded solutions

TL;DR

SILAS is a data-driven framework that discovers polynomial ODEs with guaranteed bounded solutions by jointly identifying the ODE and Lyapunov functions through polynomial optimization.

Contribution

It extends prior quadratic ODE methods to broader nonlinear systems and introduces a novel initialization for provably bounded trajectory discovery.

Findings

Successfully recovers accurate bounded ODE models for over 100 nonlinear systems.

Extends polynomial ODE discovery to a wider class with provable boundedness.

Uses a novel, model-agnostic initialization method for Lyapunov functions.

Abstract

We introduce SILAS, a data-driven framework for discovering polynomial ordinary differential equations (ODEs) with provably bounded trajectories. Boundedness is certified by compact absorbing sets defined via polynomial Lyapunov functions. We jointly identify the ODE vector field and the Lyapunov function using a well-posed nonconvex optimization problem built using polynomial optimization tools. We solve this problem using an alternating block-coordinate optimization scheme with convex subproblems, whose feasibility is ensured by a novel model-agnostic initialization that identifies a candidate Lyapunov function from data. Our methods extend prior approaches for quadratic ODEs with absorbing ellipsoids to a significantly broader class of ODEs and absorbing sets. A suite of over 100 examples demonstrates that SILAS can recover accurate and provably bounded ODE models for a broad range of nonlinear dynamical systems.