(125d) Improved Computational Tractability of Optimal Experiment Design in Pyomo

Advancements within science typically involve observing phenomena and utilizing experimental data to verify a scientific hypothesis to explain why or how this observation can be explained. These experiments require thought and experience to curate data revealing important scientific information about the hypothesis. However, in many areas of interest, experimental campaigns are limited by costs associated with material availability (e.g., novel pharmaceutical materials), labor or time availability (e.g., biological or population experiments), and equipment availability (e.g., atomic material characterization). For this reason, it is essential to design experimental campaigns that maximize the information about the system in the least number of experiments possible.

Typical experimental design procedures involve factorial [1], partial factorial, or space-filling designs [2]. However, these procedures scale with the number of experimental design decisions, presenting significant challenges with high-cost systems. Model-based design of experiments (MBDoE) utilizes a mathematical model to guide experimentation by fitting a surface to experimental data and leveraging statistical information content to design the most impactful experiments [3]. Typically, this is done with response surface models [4], which ignore any knowledge of the first-principles equations that dictate system behavior. Therefore, science-based design of experiments (SBDoE) utilizes the same statistical information criteria as MBDoE but leverages the mathematical structure of the first-principles model to guide optimal experimental design.

Over the past couple of years, we have developed and improved Pyomo.DoE [5], an open-source, equation-oriented experimental design tool to automate building and solving optimal SBDoE problems. We typically use the most common statistical criteria, A- and D-optimality, to minimize the average variance of the estimated parameters or the overall volume of the confidence ellipsoid of the estimated parameters, respectively. These criteria are convenient because they can be posed directly within an equation-oriented framework relatively quickly. However, the most popular objective, D-optimality, tends to focus on improving the confidence in the direction of the highest confidence, subsequently ignoring areas of high uncertainty within the model. Therefore, E- and ME- (or K-) optimality, minimizing the uncertainty of the most uncertain parameter or balancing the uncertainty of the most and least certain parameters, respectively, are significantly more relevant in cases where there are parameters with relatively large uncertainty.

To the author’s knowledge, these criteria have remained unimplemented in equation-oriented platforms (such as Pyomo.DoE) due to the difficulty of extracting eigenvalues on the fly within an algebraic program. However, using grey-box optimization within Pyomo [6], these objectives can be realized within Pyomo.DoE. One challenge while implementing grey-box models within Pyomo is the high-quality derivative information required of the external computation (e.g., the E- or ME-optimality objectives). Therefore, analytical expressions for E- and ME-optimality objective derivatives have been derived and implemented. In addition, computing D-optimal designs in Pyomo.DoE currently requires factorization to explicitly compute the determinant in an algebraic form [7]. With the grey-box approach, D-optimal designs have seen a speedup of two times with less than half of the solver iterations required when compared with the explicit algebraic alternative. We present these new capabilities in Pyomo.DoE and show their ability to improve models that have parameters with identifiability problems. Ongoing work includes benchmarking the new GreyBox implementation on a suite of MBDoE test problems. This includes challenging large-scale applications, such as carbon dioxide removal, to identify isotherm parameters and optimal experimental conditions for improved model quality with plans for future model (isotherm) discrimination. The addition of grey-box objectives diversifies the functionality of Pyomo.DoE and improves computational efficiency on popular objectives (e.g., D-optimality) when compared with the current algebraic implementation.

Acknowledgement

The authors acknowledge funding from the U.S. Department of Energy, Office of Fossil Energy and Carbon Management, through the Carbon Dioxide Removal Program.

Disclaimer

This project was funded by the Department of Energy, National Energy Technology Laboratory an agency of the United States Government, in part, through a support contract. Neither the United States Government nor any agency thereof, nor any of its employees, nor the support contractor, nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, or any of their contractors.

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