Data-driven discovery and control of multistable nonlinear systems and hysteresis via structured Neural ODEs

TL;DR

This paper introduces a structured Neural ODE framework for modeling and controlling multistable nonlinear systems with hysteresis, ensuring stability and capturing multiple equilibria from data.

Contribution

It proposes a novel NODE architecture that enforces stability and models multistability using a specific vector field parameterization, improving data efficiency and control.

Findings

Efficiently captures multiple basins of attraction in nonlinear benchmarks.

Enables gradient-based feedback control via the implicit equilibrium map.

Ensures trajectory stability through a contraction-enforcing structure.

Abstract

Many engineered physical processes exhibit nonlinear but asymptotically stable dynamics that converge to a finite set of equilibria determined by control inputs. Identifying such systems from data is challenging: stable dynamics provide limited excitation and model discovery is often non-unique. We propose a minimally structured Neural Ordinary Differential Equation (NODE) architecture that enforces trajectory stability and provides a tractable parameterization for multistable systems, by learning a vector field in the form $F(x,u) = f(x)\,(x - g(x,u))$, where $f(x) < 0$ elementwise ensures contraction and $g(x,u)$ determines the multi-attractor locations. Across several nonlinear benchmarks, the proposed structure is efficient on short time horizon training, captures multiple basins of attraction, and enables efficient gradient-based feedback control through the implicit equilibrium map $g$.