(125f) A Framework for Uncertainty Quantification Based on Differentiable Programming

Uncertainty quantification (UQ) in mapping/propagation is a ubiquitous problem in science. It permeates our field broadly, ranging from applications associated with the biopharmaceutical manufacturing sector [1], epidemiological modeling [2], and integrated fuel and process design [3] to name a few. Therefore, its importance cannot be underestimated. Traditional uncertainty mapping methods [4], such as Monte Carlo (MC) sampling, polynomial chaos expansion (PCE), and linear methods based on Taylor series expansions often face limitations when dealing with nonlinearities or high-dimensional problems, given inherent computational burden or model selection complexity.

Motivated by this, in this work, we developed a UQ approach that is based on differentiable programming concepts. This novel approach takes advantage of the increased commoditization of machine learning frameworks [5], especially the ones related to automatic differentiation of complete codebases, that gives access to accurate, precise and fast evaluations of high-order derivatives of implicit and high-dimensional mathematical models.

Our approach is conceptually straightforward, consisting of employing the implicit function theorem (IFT) and a path integration boundary from Domain to Image. This has been applied generally as an inverse problem [6], and in this work we motivate the boundaries in terms of uncertainty regions generated by covariance among parameters, as well as more complex uncertainty envelopes, including multimodal and high-dimensional distributions (e.g., described by ellipsoids). Our uncertainty mapping approach inherits some desired properties from our previous work [6] such as: (i) the direction of the uncertainty mapping is flexible, and uncertainties can be mapped as a forward or inverse problem which relates to uncertainty propagation and inverse UQ; (ii) it does not require brute-force enumeration to obtain the Image; and (iii) it is conceptually simpler to be employed when compared to the current uncertainty mapping techniques in the literature while maintaining adequate solution accuracy, as long as a differentiable process model is available.

Case studies related to process systems engineering applications of increased complexity are addressed, ranging from simple models with analytical solutions to systems of differential equations, in which an iterative solution is required in order to quantify uncertainty related to process variables. The results show that the proposed method when compared against Monte Carlo simulations, yields accurate results while being easier to employ and computationally more efficient. Additionally, it maps uncertainty regions within a confidence interval, irrespective of the shape of the given region. Therefore, this work can be considered as an initial step towards employing differentiable programming in a practical way for engineering problems related to UQ.

References

[1] L. H. Mervin, S. Johansson, E. Semenova, K. A. Giblin, and O. Engkvist, “Uncertainty quantification in drug design,” Drug Discov. Today, vol. 26, no. 2, pp. 474–489, Feb. 2021, doi: 10.1016/j.drudis.2020.11.027.

[2] “The impact of uncertainty on predictions of the CovidSim epidemiological code | Nature Computational Science.” Accessed: Apr. 03, 2025. [Online]. Available: https://www.nature.com/articles/s43588-021-00028-9

[3] M. Panofen, P. Ackermann, J. Viell, A. Mitsos, and M. Dahmen, “Uncertainty Quantification in Integrated Fuel and Process Design,” Energy Fuels, vol. 38, no. 15, pp. 14743–14756, Aug. 2024, doi: 10.1021/acs.energyfuels.4c02285.

[4] S. H. Lee and W. Chen, “A comparative study of uncertainty propagation methods for black-box-type problems,” Struct. Multidiscip. Optim., vol. 37, no. 3, pp. 239–253, Jan. 2009, doi: 10.1007/s00158-008-0234-7.

[5] J. R. Kitchin, V. Alves, and C. D. Laird, “Beyond the fourth paradigm of modeling in chemical engineering,” Nat. Chem. Eng., vol. 2, no. 1, pp. 11–13, Jan. 2025, doi: 10.1038/s44286-024-00170-x.

[6] V. Alves, J. R. Kitchin, and F. V. Lima, “An inverse mapping approach for process systems engineering using automatic differentiation and the implicit function theorem,” AIChE J., vol. 69, no. 9, p. e18119, 2023, doi: 10.1002/aic.18119.