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

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 $g$.