Abstract
We describe a method for the identification of models for dynamical systems from observational data. The method is based on the concept of symbolic regression and uses genetic programming to evolve a system of ordinary differential equations (ODE).
The novelty is that we add a step of gradient-based optimization of the ODE parameters. For this we calculate the sensitivities of the solution to the initial value problem (IVP) using automatic differentiation.
The proposed approach is tested on a set of 19 problem instances taken from the literature which includes datasets from simulated systems as well as datasets captured from mechanical systems. We find that gradient-based optimization of parameters improves predictive accuracy of the models. The best results are obtained when we first fit the individual equations to the numeric differences and then subsequently fine-tune the identified parameter values by fitting the IVP solution to the observed variable values.
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References
Bongard, J., Lipson, H.: Automated reverse engineering of nonlinear dynamical systems. Proc. Nat. Acad. Sci. 104(24), 9943–9948 (2007)
Chen, T.Q., Rubanova, Y., Bettencourt, J., Duvenaud, D.K.: Neural ordinary differential equations. In: Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Cesa-Bianchi, N., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 31, pp. 6571–6583. Curran Associates, Inc. (2018). http://papers.nips.cc/paper/7892-neural-ordinary-differential-equations.pdf
Gaucel, S., Keijzer, M., Lutton, E., Tonda, A.: Learning dynamical systems using standard symbolic regression. In: Nicolau, M., et al. (eds.) EuroGP 2014. LNCS, vol. 8599, pp. 25–36. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44303-3_3
Iba, H.: Inference of differential equation models by genetic programming. Inf. Sci. 178(23), 4453–4468 (2008). https://doi.org/10.1016/j.ins.2008.07.029
Isermann, R., Münchhof, M.: Identification of Dynamic Systems: An Introduction with Applications. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-540-78879-9
Kommenda, M., Kronberger, G., Winkler, S., Affenzeller, M., Wagner, S.: Effects of constant optimization by nonlinear least squares minimization in symbolic regression. In: Proceedings of the 15th Annual Conference Companion on Genetic and Evolutionary Computation, pp. 1121–1128. ACM (2013)
Schmidt, M., Lipson, H.: Data-mining dynamical systems: automated symbolic system identification for exploratory analysis. In: 9th Biennial Conference on Engineering Systems Design and Analysis, Volume 2: Automotive Systems; Bioengineering and Biomedical Technology; Computational Mechanics; Controls; Dynamical Systems, Haifa, Israel. ASME, July 2008
Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2009). https://doi.org/10.1126/science.1165893
Schmidt, M., Lipson, H.: Supporting online material for distilling free-form natrual laws from experimental data, April 2009. https://science.sciencemag.org/content/suppl/2009/04/02/324.5923.81.DC1
Topchy, A., Punch, W.F.: Faster genetic programming based on local gradient search of numeric leaf values. In: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation, pp. 155–162. Morgan Kaufmann Publishers Inc. (2001)
Worm, T., Chiu, K.: Scaling up prioritized grammar enumeration for scientific discovery in the cloud. In: IEEE International Conference on Big Data, pp. 621–626. IEEE (2014)
Acknowledgments
The authors gratefully acknowledge support by the Austrian Research Promotion Agency (FFG) within project #867202, as well as the Christian Doppler Research Association and the Federal Ministry of Digital and Economic Affairs within the Josef Ressel Centre for Symbolic Regression.
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Kronberger, G., Kammerer, L., Kommenda, M. (2020). Identification of Dynamical Systems Using Symbolic Regression. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2019. EUROCAST 2019. Lecture Notes in Computer Science(), vol 12013. Springer, Cham. https://doi.org/10.1007/978-3-030-45093-9_45
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DOI: https://doi.org/10.1007/978-3-030-45093-9_45
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