Hybrid models are an amalgamation of first-principles models and data-driven models. Hybrid models’ success is accredited to their intended structure of overcoming the shortcomings of first principles models (like high computation expense, uncertainties in parameter estimation) and data-driven models (like overfitting, no physical significance, extrapolation capabilities). Hybrid models are more adaptive than first-principles models and can handle uncertainties efficiently. Since its inception, hybrid models are being used in several domains of engineering to solve complex systems, especially in biological systems for modeling, simulation, and optimization [1, 2]. While the hybrid models have several advantages, it is necessary to employ a good strategy while designing hybrid models (e.g., series integration, parallel integration). A good strategy would be to establish an understanding of the significance of all parameters in the hybrid model to justify the choice of the user-dependent data-driven model. Otherwise, the resulting hybrid model would only perform well locally with no global-performance guarantees. While parameter analysis techniques have been implemented for first-principles models, parameter analysis for data-driven models is relatively an upcoming domain of research. Hence, the analysis of hybrid models has not been extensively studied.

Sensitivity analysis (SA) has been historically used to understand models, their important parameters, and for validation purposes. Sensitivity (or what-if) analysis refers to the study of changes in the output of a model due to large or extreme changes in the input [3]. Global sensitivity analysis methods have been proposed to study the variation in inputs over the input domain and associate it with the output uncertainties [4]. Functional ANOVA (FANOVA) is an SA tool that models the multivariate functions as a sum of their individual input effects and interactions and has been commonly used for first-principles models [5]. On the other hand, data-driven models are mainly optimized and analyzed through Bayesian optimization; a Gaussian process-based global optimization. Bayesian Optimization is well-suited for handling computationally expensive objective functions as it uses the acquisition function for optimization [6]. Given there is no proposed common ground for sensitivity analysis of first-principles and data-driven models, we extend sensitivity analysis studies to hybrid models. In our proposed work, we plan to decompose the hybrid model into Koopman eigenfunctions as they capture nonlinear dynamics in the form of linear models [7]. The linear model will be optimized using a Bayesian Optimization-based Dynamic Mode Decomposition method. The sensitivity of parameters will be finally analyzed using FANOVA through Sobol’s indices estimation. This work will provide insights into the impact of chosen black box model on the overall hybrid model. The advantage of this approach is demonstrated through application to isothermal CSTR, bioreactor, and hydraulic fracturing systems.

References

[1] P. Shah, M. Sheriff, M. Bangi, C. Kravaris, J. Kwon, C. Botre and J. Hirota, "Deep neural network-based hybrid modeling and experimental validation for an industry-scale fermentation process: Identification of time-varying dependencies among parameters," Chemical Engineering Journal, vol. 441, no. 1, p. 135643, 2022.

[2] S. Zendehboudi, N. Rezaei, and A. Lohi, "Applications of hybrid models in chemical, petroleum, and energy systems: A systematic review," Applied Energy, vol. 228, no. 1, pp. 2539-2566, 2018.

[3] J. Kleijnen, "Sensitivity Analysis versus Uncertainty Analysis: When to Use What?," in Predictability and Nonlinear Modelling in Natural Sciences and Economics, Springer, Dordrecht, 1994, pp. 322-333.

[4] A. Saltelli, K. Chan, E. Scott and editors, Sensitivity Analysis, Wiley, 2009.

[5] R. Ghanem, D. Higdon, and H. Owhadi, Handbook of Uncertainty Quantification, Springer International, 2017.

[6] M. Gelbart, J. Snoek and R. Adams, "Bayesian Optimization with Unknown Constraints," 22 Mar 2014. [Online]. Available: https://arxiv.org/abs/1403.5607. [Accessed 9 April 2022].

[7] E. Kaiser, J. Kutz, and S. Brunton, "Data-Driven Approximations of Dynamical Systems Operators for Control," in The Koopman Operator in Systems and Controls, A. Mauroy, I. Meźic, and Y. Susuki, Eds., Springer Nature, 2020, pp. 197-234.