Hybrid modeling and prediction of dynamical systems
Fig 2
Comparison of the prediction methods in the Lorenz-63 system.
Results of predicting the Lorenz-63 (a) x, (b) y and (c) z variables averaged over 500 realizations. Error bars denote standard error over the 500 realizations. Training data consists of 500 data points generated from Eq 6 with σ = 10, ρ = 28 and β = 8/3 with sample rate h = 0.05. Data are corrupted by Gaussian observational noise with mean 0 and variance of 4. Parametric (black), nonparametric (blue) and hybrid (red) prediction accuracy with parameter uncertainty of 80% (solid line) plotted as a function of forecast horizon. Hybrid prediction, which utilizes mechanistic equations in describing x and z but nonparametrically represents y, offers an improvement in short-term prediction accuracy over standalone nonparametric prediction. Parametric prediction at this uncertainty level performs poorly in predicting all three variables and in the case of (b) and (c) is not seen due to the scale of the error. For comparison purposes, the parametric method with 50% (dotted line) and 20% (dashed line) uncertainty is also considered. As the uncertainty shrinks, performance of the parametric method improves. However, only at a small uncertainty level does it outperform the short-term improvement in prediction afforded by the hybrid method.