Gaussian process learning with flow map refinement for parameter estimation in dynamical systems
Authors: Yue Hao, Dongwei Ye
Organizations: School of Mathematical Science, Suzhou University of Science and Technology, Suzhou, PR China · Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, PR China
Parameter estimation is a central task in data-driven learning of dynamical systems. It aims to recover the underlying physical parameters from observed time-series data, thereby providing interpretable insights into the physical mechanisms governing the system. Gradient/derivative matching methods based on Gaussian process provide an efficient way to perform parameter estimation. Those methods avoid repeated numerical integration and enforce local derivative consistency. However, such local matching may result in global inconsistency with the governing flow map, particularly under scarce and noisy observations. To address this limitation, we propose a framework based on Gaussian process learning with flow map refinement (GPL-FMR), a two-stage parameter estimation framework. The first stage is based on Gaussian process learning algorithm and the posterior obtained from which is transferred as an informative prior to the second stage based on flow-map refinement. The second stage further improves the parameter estimation via optimisation based on global dynamical constraints. We demonstrate and analyse its performance on multiple numerical examples, including the Van der Pol oscillator, the Lotka-Volterra model, and the Lorenz-63 system. The results show that the proposed framework consistently improves parameter estimation accuracy, particularly under scarce and noisy observations.
Figures & tables
Figure 1: Schematic diagram of the proposed two-stage parameter estimation framework. The first stage employs Gaussian process learning (blue), while the second stage applies flow-map refinement (red) to further refine the parameter estimates.
Figure 2: Trajectory reconstruction for the Van der Pol oscillator using the estimated parameters and the prescribed initial condition, with Nt=20 and Nt=100 observations under 10% and 30% noise levels. The uncertainty bands correspond to two standard deviations obtained from the Laplace approximation.
Figure 3: Comparison between the trajectories reconstructed from the inferred latent states and the GP-smoothed trajectories obtained from the GPL stage for the Van der Pol oscillator under 30% noise.
Figure 4: Comparison of the relative errors in parameter estimation for the Lotka-Volterra model using feature regression (FR), GPL, and the proposed GPL-FMR method, with 20,30,40,50, and 100 observations under 10%, 20%, and 30% noise levels.
Figure 5: Trajectory reconstruction for the Lotka-Volterra model using the estimated parameters and prescribed initial condition, with Nt=20 and Nt=100 observations under 10% and 30% noise levels. The uncertainty bands correspond to two standard deviations obtained from the Laplace approximation.
Figure 6: Comparison between the trajectories reconstructed from the inferred latent states and the GP-smoothed trajectories obtained from the GPL stage for the Lotka-Volterra model under 30% noise.
Figure 7: Comparison of the relative errors in parameter estimation for the Lotka-Volterra model using feature regression (FR), GPL, and the proposed GPL-FMR method, with 20,30,40,50, and 100 observations under 10%, 20%, and 30% noise levels.
Figure 8: Trajectory reconstruction for the Lorenz-63 model using the estimated parameters and prescribed initial condition, with Nt=20 and Nt=100 observations under 10% noise levels. The uncertainty bands correspond to two standard deviations obtained from the Laplace approximation.
Figure 9: Trajectory reconstruction for the Lorenz-63 model using the estimated parameters and prescribed initial condition, with Nt=20 and Nt=100 observations under 30% noise levels. The uncertainty bands correspond to two standard deviations obtained from the Laplace approximation.
Figure 10: Comparison between the trajectories reconstructed from the inferred latent states and the GP-smoothed trajectories obtained from the GPL stage for the Lorenz-63 system under 30% noise.