(147ai) Measure This, Not That: Pareto Optimal Trade-Offs between Model-Based Information Content and Measurements Budget

Model-based Design of Experiments (MBDoE) is a powerful tool for building and validating mathematical models. It leverages science-based models to maximize information gain, represented by Fisher Information Matrix (FIM), from experiments while minimizing time and resource costs by optimizing experimental conditions such as temperature, pressure, or batch time [1]. FIM is evaluated by the dynamical sensitivities of measurements. When insufficient or unsuitable measurements are collected, uncertainty parameters in the science-based mathematical model may remain unidentifiable or MBDoE may suggest uninformative or infeasible optimal experiments [2].

In this work, we propose a novel convex optimization formulation to compute the best set of measurements for multi-response dynamical systems with asynchronous time steps that maximize the Fisher information content subject to budget constraints. The trace (A-optimality) or determinant (D-optimality) of the Fisher Information Matrix (FIM) quantifies the information content. The framework supports arbitrary (positive semi-definite) variance and covariances between every pair of responses and their time steps.

To solve this problem with less computational burden, we leveraged the continuous-effort design (CED) concept [3] where the binary decisions are relaxed to be continuous. This continuous design allows the formulation of a convex problem, since the classical design criterion, A- and D-optimality yield a concave function in the measurement selectors. This way, local sub-optimal designs can be avoided because the relaxed CED problem is convex and can be efficiently solved with modern convex optimization algorithms. An obvious drawback of the CED method is that the size of the optimization problem scales with the number of measurements, But the sensitivity matrix can be precomputed prior to solving the optimization problem, which reduces the computational burden. In this work, all process models are constructed on Pyomo [4] and the precomputed sensitivity matrix is obtained by Pyomo.DoE [2]. Another numerical challenge is that although the determinant of FIM is a concave function, the evaluation equations of the determinant in Pyomo sabotage the convexity of this problem. We discuss strategies to use the new GreyBox modeling feature in Pyomo to efficiently handle the calculation of the log determinant of the FIM.

We first illustrate the methodology with a reaction kinetics example, a multi-response system with different variances and covariances between measurements. We find that the continuous relaxation quickly identifies a discrete global optimum. Next, to show the scalability of the method, we consider a rotary-bed CO2 adsorption system, modeling with partial differential-algebraic questions (PDE). This system has at least 14 candidate measurements each with up to 220 time steps. We show the flexibility of the framework by classifying these measurements as static-cost or dynamic-cost. Static-cost measurements take a one-time installation cost and have no extra costs for the measurement at each time step. In contrast, dynamic-cost measurements require both an installation cost as well as an extra cost for measuring each time step (e.g., a laboratory technician takes samples manually). To show the flexibility of the framework, we add practical constraints, including requiring at least 10 minutes between each dynamic-cost measurement, constraining the total number of samples from all dynamic-cost measurements, and the total number for each dynamic measurements. The optimization problem has up to 157,663 variables, 17 equality constraints, and 472,347 inequality constraints. The relaxed problem is formed with A-optimality as the objective function, solved by IPOPT [5] in minutes, and formed with D- optimality as the objective function, where the log determinant objective function is evaluated by grey-box block, solved by cyipopt. The mixed-integer problem is also solved to round solutions, where A-optimality problem is solved by Gurobi [6], and D-optimality problem is solved by mindtpy.

In conclusion, we find that the method can quickly solve the global optimum in the relaxed measurement space and identify a solution where most decisions are discrete without needing rounding. The continuous-effort design concept and the use of grey-box module retains the convexity of the optimization problem regardless of model structure. With precomputed sensitivity matrix from Pyomo.DoE, the optimization problem can be scaled to more measurements with less computational burden. The mixed-integer solution is also solved to round solutions. Using this approach, we character the Pareto tradeoff between measurement costs and measurement values, which provides valuable insights into the physical system.

Reference

[1] Franceschini, G., & Macchietto, S. (2008). Model-based design of experiments for parameter precision: State of the art. Chemical Engineering Science, 63(19), 4846-4872.

[2] Wang, J., & Dowling, A. W. (2022). Pyomo. DOE: An open‐source package for model‐based design of experiments in Python. AIChE Journal, 68(12), e17813.

[3] Kusumo, K. P., Kuriyan, K., Vaidyaraman, S., García-Muñoz, S., Shah, N., & Chachuat, B. (2022). Risk mitigation in model-based experiment design: a continuous-effort approach to optimal campaigns. Computers & Chemical Engineering, 159, 107680.

[4] Bynum, M. L., Hackebeil, G. A., Hart, W. E., Laird, C. D., Nicholson, B. L., Siirola, J. D., ... & Woodruff, D. L. (2021). Pyomo-optimization modeling in python (Vol. 67). Berlin/Heidelberg, Germany: Springer.

[5] Biegler, L. T., & Zavala, V. M. (2009). Large-scale nonlinear programming using IPOPT: An integrating framework for enterprise-wide dynamic optimization. Computers & Chemical Engineering, 33(3), 575-582.

[6] Gurobi Optimization LLC (2023). Gurobi Optimizer Reference Manual. https://www.gurobi.com.