A Bi-Stage Gaussian Process Framework for Modeling and Predicting Nonlinear Non-Autonomous Dynamics

Learning nonparametric systems of Ordinary Differential Equations (ODEs) from noisy data is challenging, especially when the system is input-dependent. Most current nonparametric approaches focus on autonomous systems, making them unable to capture the influence of external inputs. In this paper, we introduce a Bi-stage Gaussian Process (GP) framework for non-autonomous ODEs, capable of estimating system states and their derivatives directly from noisy measurements. The proposed method adopts a purely data-driven and nonparametric formulation, relying on Gaussian process regression and numerical integration without assuming explicit parametric system models or theoretical performance guarantees. The method is demonstrated on a scalar forced ODE with amplitudes A ∈ [0.05, 2.5]π and frequencies ω ∈ [0.1, 31.6], achieving state prediction errors below 2% for high signal-to-noise ratios (SNR=1000) and derivative errors below 5% even for noisy measurements (SNR=30). Furthermore, the approach is applied to a continuous stirred tank reactor (CSTR) system with inlet concentrations C_A0 = 1.0 2.0 mol/m³ and flow rates F = 0.01 m³/s, successfully estimating reaction rates with relative errors below 4% across varying noise levels (SNR=100 30). Comparative results with non-parametric ODE (npODE), Gaussian Process ODE (GPODE) and continuous-time state-space neural network (CSNN) models demonstrate that the proposed Bi-stage GP achieves superior generalization performance under varying input conditions. The results demonstrate that the proposed method is robust, accurate, and capable of generalizing to unobserved inputs, providing a reliable alternative to classical ODE modeling in noisy and complex systems.