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

Abstract

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 <inline-formula> <tex-math notation="LaTeX">$A \in [{0.05, 2.5}]\pi $ </tex-math></inline-formula> and frequencies <inline-formula> <tex-math notation="LaTeX">$\omega \in [{0.1, 31.6}]$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$C_{A0}=1.0 2.0$ </tex-math></inline-formula> mol/m3 and flow rates <inline-formula> <tex-math notation="LaTeX">$F=0.01$ </tex-math></inline-formula> m3/s, successfully estimating reaction rates with relative errors below 4% across varying noise levels (SNR <inline-formula> <tex-math notation="LaTeX">$=100~30$ </tex-math></inline-formula>). 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.