A Hybrid Symbolic Regression Method for the Problem of Dynamic System Identification

Identifying dynamic systems from data is a complex problem, where a key requirement of modern research is not only accuracy but also model interpretability. Although highly effective, the symbolic regression method based on genetic programming has inherent limitations, the most important of which is stochasticity, leading to instability of results. In this paper, a new hybrid method, GP-SINDy, is proposed to overcome these shortcomings. Its core idea is to combine two approaches: genetic programming performs a global search for the model structure, while sparse identification fine-tunes the corresponding parameters. The effectiveness of the proposed method was validated through comprehensive computational experiments. On test data, GP-SINDy demonstrated the ability to find models with an optimal balance of accuracy and complexity, outperforming the baseline genetic programming algorithm. Analysis on noisy data confirmed the increased efficiency of the proposed method. Verification on a real system demonstrated the practical applicability of the approach for constructing adequate analytical models. Thus, the GP-SINDy hybrid method represents a powerful and versatile tool for automatically deriving interpretable system dynamics equations, opening up new possibilities in various fields of science and engineering.

1. Bakhtadze N. N., Ginsberg K. S., Borovskikh L. P. Identification of systems on the way to creating a general theory of identification of control objects, Problems of control, 2015, no. 3, pp. 79—83 (in Russian).

2. Udrescu S., Tegmark M. Ai feynman: a physics-inspired method for symbolic regression, Science Advances, 2020, vol. 6, no. 16, DOI: 0.1126/sciadv.aay2631.

3. Schmidt M., Lipson Н. Distilling free-form natural laws from experimental data, Science, 2009, vol. 324, no. 5923, pp. 81—85.

4. Makke N., Chawla S. Interpretable scientific discovery with symbolic regression: a review, Artif Intell Rev, 2024, vol. 57, no. 2, DOI: 10.1007/s10462-023-10622-0.

5. Kabliman E., Kolody A. H., Kommenda M., Kronberger G. Prediction of stress-strain curves for aluminium alloys using symbolic regression, AIP Conf Proc, 2019, vol. 2113, no. 1, DOI: 10.1063/1.5112747.

6. Koza J. R. Genetic programming: On the programming of computers by means of natural selection, The MIT Press, 1992, 836 p.

7. Poli R., Langdon W. B., McPhee N. F. А field guide to genetic programming, with contributions by J. R. Koza, GPBiB, 2008, 252 p.

8. Iba H., Sakamoto E. Inference of Differential Equation Models by Genetic Programming, Information Sciences, 2008, no. 178, no. 23, pp. 4453—4468.

9. Ljung L. System identification: theory for the user, Englewond Cliffs, Prentice Hall, 1999.

10. Kumpati S. N., Kannan P. Identification and control of dynamical systems using neural networks, IEEE Transactions on neural networks, 1990, vol. 1, no. 1, pp. 4—27.

11. Rudin C. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead, Nat. Mach. Intell, 2019, vol. 1, no. 5, pp. 206—215.

12. Ni H., Zeng F., Yu B., Sun F. The convergence mechanism and strategies for non-elitist genetic programming, Applied Mechanics and Materials, 2013, pp. 3850—3860.

13. Zhang L. Analysis of the trade-off between the accuracy and complexity of identified models of dynamic systems, Data analysis and processing systems, 2024, vol. 2, no. 94, pp. 85—93 (in Russian).

14. Zhang Н, Zhou A. RL-GEP: symbolic regression via gene expression programming and reinforcement learning, 2021 International Joint Conference on Neural Networks, 2021, pp. 1—8.

15. Carlos A., Moshe S. Fuzzy CoCo: А cooperative-coevolutionary approach to fuzzy modeling, IEEE transactions on fuzzy systems, 2001, vol. 9, no. 5, pp. 727—737.

16. Zhang L. А hybrid method for structural-parameter identification of dynamic systems, Data analysis and processing systems, 2025, vol. 2, no. 98, pp. 111—122 (in Russian)

17. Zhang L., Filimonov N. B. Identification of dynamic systems based on experimental data processing using genetic programming, Journal of advanced research in natural science, 2023, no. 18, pp. 2—12 (in Russian).

18. Brunton S., Proctor J., Kutz J. Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proceedings of the National Academy of Sciences, 2016, vol. 113, no. 15, pp. 3932—3937.

19. Brunton S., Kutz J. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control, Cambridge: Cambridge University Press, 2019, 495 p.

20. Murphy H., Jaafari H., Dobrovolny Н. M. Differences in predictions of ODE models of tumor growth: a cautionary example, International Journal of Vehicle Noise and Vibration, 2015, vol. 11, no. 1, pp. 78—101.