(13a) Reduced-Space Multi-Fidelity Bayesian Optimization of Process Flowsheets

Flowsheet models are integral to process systems engineering, enabling the simulation and optimization of complex (bio)chemical processes. These models offer insights into key performance indicators (KPIs) such as cost, efficiency, and environmental impact. However, these models are often treated as black boxes, particularly when using commercial simulation software that relies on steady-state assumptions and heuristic rules. In such cases, the underlying equations and derivative information are inaccessible, making conventional gradient-based optimization approaches impractical. Data-driven optimization techniques, such as Bayesian Optimization (BO), are well-suited for such expensive and derivative-free problems [1,2].

While BO is well-suited for low-sample, black-box problems, its scalability to high-dimensional settings remains a known challenge [3]. To address this, Global Sensitivity Analysis (GSA) [4] can be employed to identify influential decision variables that drive variability in the objective function and reduce problem dimensionality. By quantifying both individual and interaction effects [5], GSA can help streamline the optimization process by focusing on the most impactful inputs. While this enables more efficient exploration, BO remains computationally expensive when applied directly to high-fidelity simulation models. Conversely, surrogate-based approaches using feedforward artificial neural networks (ANNs) can drastically reduce evaluation time, but suffer from model mismatch: their predicted objective values often deviate from the true objective function, potentially yielding suboptimal solutions when used in isolation [6].

To address this trade-off, we propose a multi-fidelity Bayesian optimization (MFBO) [7] framework that integrates both high-fidelity simulator evaluations and low-fidelity ANN surrogates. The key insight underpinning this approach is that, although ANN predictions may be biased or imprecise in terms of objective value, they nonetheless provide informative guidance toward promising regions in the decision space. We model the high-fidelity objective as a linear transformation of the low-fidelity function, augmented by a discrepancy term, i.e.,

ƒ_high(x) = ρ·ƒ_low(x) + δ(x),

where ρ is a scaling parameter and δ(x) is a zero-mean Gaussian process capturing the residual. These parameters are estimated via maximum likelihood using co-located evaluations from both fidelities. A multi-fidelity Gaussian process model is constructed over this joint space, enabling the use of acquisition functions such as multi-fidelity expected improvement (MF-EI) to select optimal inputs and fidelity levels at each iteration.

We evaluate the proposed MFBO framework on an industrially relevant case study: plasmid DNA production modeled in SuperPro Designer with 18 decision variables. Comparisons are drawn against baseline strategies including simulation-based BO (with and without GSA) and surrogate-based BO. Results demonstrate that MFBO achieves near-optimal solutions while reducing total computational cost by up to two orders of magnitude relative to high-fidelity simulation-based BO. Moreover, MFBO exhibits improved reliability over surrogate-only strategies, by explicitly accounting for surrogate bias and uncertainty. These findings underscore the utility of multi-fidelity optimization for real-time or resource-constrained flowsheet design problems.

References :

[1] F. Boukouvala, M.G. Ierapetritou, "Derivative-free optimization for expensive constrained problems using a novel expected improvement objective function," AIChE J., 60(7):2462–2474, 2014.

[2]. D. van de Berg, T. Savage, P. Petsagkourakis, D. Zhang, N. Shah, E.A. del Rio-Chanona, "Data-driven optimization for process systems engineering applications," Chem. Eng. Sci., 248:117135, 2022.

[3] E. Brochu, V.M. Cora, N. De Freitas, "A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning," arXiv preprint, arXiv:1012.2599, 2010.

[4] A. Saltelli, P. Annoni, I. Azzini, F. Campolongo, M. Ratto, S. Tarantola, "Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index," Comput. Phys. Commun., 181(2):259–270, 2010.

[5] S. Kucherenko, "SobolHDMR: a general-purpose modeling software," Synth. Biol., pp. 191–224, 2013.

[6] N. Triantafyllou, B. Lyons, A. Bernardi, B. Chachuat, C. Kontoravdi, M.M. Papathanasiou, "Comparative assessment of simulation-based and surrogate-based approaches to flowsheet optimization using dimensionality reduction," Comput. Chem. Eng., 189:108807, 2024.

[7] K. Kandasamy, G. Dasarathy, J. Schneider, B. Póczos, "Multi-fidelity Bayesian optimisation with continuous approximations," in Proc. 34th Int. Conf. Mach. Learn. (ICML), PMLR 70:1799–1808, 2017.