Curriculum Multiple Shooting for Robust Training of Neural and Universal Differential Equations

Abstract

Neural ordinary differential equations (NODEs) and universal differential equations (UDEs) provide flexible and popular frameworks for learning interpretable dynamical systems from noisy time-series data. However, training these models remains challenging, and versatile methods that robustly handle sparse and noisy data as well as partially observed models are lacking. To address this, we introduce curriculum multiple shooting (CMS), a general-purpose training strategy for fitting ordinary differential equation (ODE) models to time-series data by integrating curriculum learning with multiple shooting. Across twelve benchmarks spanning simulated and real data, and covering NODEs, UDEs, and mechanistic ODEs, CMS accelerates and stabilises training convergence, outperforms state-of-the-art training strategies, and ranks among the best methods in generalisation. Finally, we discuss possible explanations for the strong performance of CMS in light of contemporary theories of what makes training on time-series challenging.