(182g) Decision-Making Under Epistemic Uncertainty Using Bayesian Hybrid Models

Predictive models are fundamental to Process Systems Engineering^[1,2]. However, all models are prone to uncertainty leading to under- or over-confident predictions which poses a challenge in a decision-making process. Aleatory uncertainty is a consequence of random phenomena such as experimental variability and noisy observations and is well studied in process engineering^[3,4]. On the other hand, epistemic or model form uncertainty arises from model inadequacies due to simplifying assumptions or neglected control variables^[5]. Moreover, the use of reduced order, surrogate, or scale-bridging models^[6,7] in multiscale frameworks also leads to information loss causing epistemic uncertainty. The effects of epistemic uncertainty are actively being studied^[8], however, the burden of correcting for epistemic uncertainty is often accomplished using strategies to improve the estimates of the mechanistic model parameters^[9]. The advent of data-driven techniques has seen the development of several hybrid modeling techniques^[10,11]. For example, in neural differential equations neural networks are introduced into the RHS of a differential equation and are trained with mechanistic constraints^[12]. Most of these techniques have shown to be effective in utilizing limited data to make accurate predictions, however, quantifying epistemic uncertainty and propagating its effect across scales remains a challenge.

Kennedy and O’Hagan proposed a statistical framework to simultaneously quantify aleatory and epistemic uncertainty^[13]. We extend this framework to develop Bayesian hybrid models of the form^[14]:

y = η(.) + δ(.) + ε

In the above equation, the glass-box component η(.) is the mechanistic model which is prone to systematic bias due to epistemic uncertainty. The black-box component δ(.) is the Gaussian Process (GP) discrepancy function which leverages data to offset the systematic bias in predictions. ε is normally distributed random noise and models aleatory uncertainty. We highlight that the proposed model lies at the intersection of glass-box models, developed from foundational scientific theory, and surrogate models which utilize statistics and machine learning to reveal trends in voluminous data. We use the GP discrepancy function as its probabilistic nature provides readily interpretable uncertainty information along with predictions. Moreover, it allows the use of Bayesian model calibration to train the model^[15]. The posterior distribution of the mechanistic model parameters and GP hyperparameters is used in a single stage stochastic program to design an optimal experiment despite epistemic uncertainty thereby completing the Bayesian decision-making workflow.

In this work, we seek to answer 3 key questions:

1. Can Bayesian hybrid models overcome epistemic uncertainty?

2. What is the optimal training strategy for Bayesian hybrid models?

3. How does the Bayesian workflow inform reaction engineering studies?

Using ballistic firing as an illustrative case study ^[14], we show that Bayesian hybrid models overcome epistemic uncertainty to hit a target 100 meters away using only 6 data points, despite using a mechanistic model that neglects air resistance effects. Moreover, we highlight that the simultaneous calibration of the glass-box and black-box components using maximum-a-posteriori point estimates is sufficient to ensure direct hits. We compare this training method with its alternatives such as fully Bayesian training and the sequential training of the individual hybrid model components. Finally, using a case study of reaction engineering, we show how this workflow can be used to learn unknown reaction kinetics and accurately predict the duration and temperature of a reaction, thus highlighting its potential in fields such as multiscale reactor design and flowsheet optimization. We conclude with remarks about how this workflow fits into our integrated molecules-to-systems engineering design framework^[16].

References:

[1] Adjiman, C. S., Sahinidis, N. V., Vlachos, D. G., Bakshi, B., Maravelias, C. T., & Georgakis, C. (2021). Process Systems Engineering Perspective on the Design of Materials and Molecules. Industrial & Engineering Chemistry Research.

[2] Tian, Y., Demirel, S. E., Hasan, M. M. F., & Pistikopoulos, E. N. An overview of process systems engineering approaches for process intensification: State of the art. Chemical Engineering and Processing, 133, 160–210 (2018).

[3] Kalyanaraman, J., Fan, Y., Labreche, Y., Lively, R. P., Kawajiri, Y. & Realff, M. J. Bayesian estimation of parametric uncertainties, quantification and reduction using optimal design of experiments for CO₂ adsorption on amine sorbents. Computers & Chemical Engineering 81, 376-388 (2015).

[4] Ruppen, D., Benthack, C. & Bonvin, D. Optimization of batch reactor operation under parametric uncertainty-computational aspects. Journal of Process Control 5, 235-240 (1995).

[5] McClarren, G. R., Uncertainty Quantification and Predictive Computational Science, A Foundation for Physical Scientists and Engineers (2018).

[6] Biegler, L. T., Lang, Y.-d. & Lin, W. Multi-scale optimization for process systems engineering. Computers and Chemical Engineering 60, 17-30 (2014)

[7] Boukouvala, F. & Floudas, C. A. Argonaut: Algorithms for global optimization of constrained grey-box computational problems. Optimization Letters 11, 895-913 (2017).

[8] Lucia, S., Andersson, J. A. ., Brandt, H., Diehl, M., & Engell, S. Handling uncertainty in economic nonlinear model predictive control: A comparative case study. Journal of Process Control, 24(8), 1247–1259 (2014).

[9] Babazadeh, R., Ghaderi, H., & Pishvaee, M. S. A benders-local branching algorithm for second-generation biodiesel supply chain network design under epistemic uncertainty. Computers & Chemical Engineering, 124, 364–380 (2019).

[10] Von Stosch, M., Oliveira, R., Peres, J., & Feyo de Azevedo, S. Hybrid semi-parametric modeling in process systems engineering: Past, present and future. Computers & Chemical Engineering, 60(C), 86–101 (2014).

[11] Ghosh, D., Moreira, J., & Mhaskar, P. Model Predictive Control Embedding a Parallel Hybrid Modeling Strategy. Industrial & Engineering Chemistry Research, 60(6), 2547-2562 (2021).

[12] De Jaegher, B., De Schepper, W., Verliefde, A., & Nopens, I. Enhancing mechanistic models with neural differential equations to predict electrodialysis fouling. Separation and Purification Technology, 259, 118028 (2021).

[13] Kennedy, M. C. & O'Hagan, A. Bayesian calibration of computer models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63, 425-464 (2001).

[14] Eugene, E. A., Gao, X. & Dowling, A. W. Learning and optimization with Bayesian hybrid models. In Proceedings of the 2020 American Controls Conference (2020).

[15] Higdon, D., Kennedy, M., Cavendish, J. C., Cafeo, J. A. & Ryne, R. D. Combining field data and computer simulations for calibration and prediction. SIAM Journal on Scientific Computing 26, 448-466 (2004).

[16] Eugene, E., Phillip, W., Dowling, A. (2019). Data Science-Enabled Molecular-to-Systems Engineering for Sustainable Water Treatment, Current Opinion in Chemical Engineering, 26, 122–130.