A Mathematical Model for Predicting Complex Dynamic Systems Using Hybrid Computational Approaches

George Em Karniadakis (Division of Applied Mathematics and School of Engineering, Brown University, Providence, Rhode Island 02912, USA)
Elena Torfs (Department of Data Analysis and Mathematical Modelling, Faculty of Bioscience Engineering, Ghent University)
Luca Marchetti (Laboratory of Computational Modeling, Department of Cellular, Computational and Integrative Biology, University of Trento)

GSE-journal
Vol. 3 No. 5 (2026): Vol. 3 No. 5 (2026): Vol. 3 No. 5 (2026): July 2026

Hybrid computational models can improve prediction when governing equations are incomplete, but their advantages are often evaluated on isolated systems and without simultaneous assessment of accuracy, stability, interpretability, and uncertainty. This study develops a modular hybrid mathematical model that combines a partially specified ordinary differential equation, a regularized neural residual, joint parameter calibration, physical constraints, and ensemble-based uncertainty quantification. The framework was evaluated through in silico experiments on five benchmark systems representing periodic, chaotic, stiff, ecological, and engineering dynamics: Van der Pol, Lorenz-63, Robertson kinetics, Lotka–Volterra, and a continuous stirred-tank reactor. Six hundred trajectories were generated using space-filling sampling, partitioned at the trajectory level, and tested under interpolation, extrapolation, measurement noise, data scarcity, and partial observability. The proposed model was compared with an incomplete mechanistic model, a neural ordinary differential equation, and sequential residual correction. In illustrative synthetic results, the hybrid model achieved a mean normalized root-mean-square error of 0.0678, reducing error by 37.3% relative to the strongest baseline. It also increased the mean time to divergence to 83.6% of the forecast horizon, reduced median mechanistic parameter error to 4.8%, limited physical-constraint violations to 0.6%, and attained 0.947 coverage for nominal 95% prediction intervals. Friedman and Holm-adjusted Wilcoxon tests indicated significant paired improvements with large effect sizes. Ablation analyses showed that residual regularization, physical constraints, and joint calibration each contributed materially. These findings illustrate how restricted data-driven correction can enhance heterogeneous dynamical-system prediction while preserving mechanistic meaning, although empirical execution and external validation are required before scientific claims are made