An official website of the United States government
Hybrid (i.e., grey-box) models are a powerful and flexible paradigm for predictive science and engineering. Grey-box models use data-driven constructs to incorporate unknown or computationally intractable phenomena into glass-box mechanistic models. The pioneering work of statisticians Kennedy and O’Hagan introduced a new paradigm to quantify epistemic (i.e., model-form) uncertainty. While popular in several engineering disciplines, prior work using Kennedy–O’Hagan hybrid models focuses on prediction with accurate uncertainty estimates. This work demonstrates computational strategies to deploy Bayesian hybrid models for optimization under uncertainty. Specifically, the posterior distributions of Bayesian hybrid models provide a principled uncertainty set for stochastic programming, chance-constrained optimization, or robust optimization. Through two illustrative case studies, we demonstrate the efficacy of hybrid models, composed of a structurally inadequate glass-box model and Gaussian process bias correction term, for decision-making using limited training data. From these case studies, we develop recommended best practices and explore the trade-offs between different hybrid model architectures.
more »
« less
This content will become publicly available on January 1, 2027
BONSAI: Structure-exploiting robust Bayesian optimization for networked black-box systems under uncertainty
Optimal design under uncertainty remains a fundamental challenge in advancing reliable, next-generation process systems. Robust optimization (RO) offers a principled approach by safeguarding against worst-case scenarios across a range of uncertain parameters. However, traditional RO methods typically require known problem structure, which limits their applicability to high-fidelity simulation environments. To overcome these limitations, recent work has explored robust Bayesian optimization (RBO) as a flexible alternative that can accommodate expensive, black-box objectives. Existing RBO methods, however, generally ignore available structural information and struggle to scale to high-dimensional settings. In this work, we introduce BONSAI (Bayesian Optimization of Network Systems under uncertAInty), a new RBO framework that leverages partial structural knowledge commonly available in simulation-based models. Instead of treating the objective as a monolithic black box, BONSAI represents it as a directed graph of interconnected white- and black-box components, allowing the algorithm to utilize intermediate information within the optimization process. We further propose a scalable Thompson sampling-based acquisition function tailored to the structured RO setting, which can be efficiently optimized using gradient-based methods. We evaluate BONSAI across a diverse set of synthetic and real-world case studies, including applications in process systems engineering. Compared to existing simulation-based RO algorithms, BONSAI consistently delivers more sample-efficient and higher-quality robust solutions, highlighting its practical advantages for uncertainty-aware design in complex engineering systems.
more »
« less
- Award ID(s):
- 2237616
- PAR ID:
- 10657426
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Computers & Chemical Engineering
- Volume:
- 204
- Issue:
- C
- ISSN:
- 0098-1354
- Page Range / eLocation ID:
- 109393
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Bayesian optimization is a sample-efficient black-box optimization procedure that is typically applied to a small number of independent objectives. However, in practice we often wish to optimize objectives defined over many correlated outcomes (or “tasks”). For example, scientists may want to optimize the coverage of a cell tower network across a dense grid of locations. Similarly, engineers may seek to balance the performance of a robot across dozens of different environments via constrained or robust optimization. However, the Gaussian Process (GP) models typically used as probabilistic surrogates for multi-task Bayesian optimization scale poorly with the number of outcomes, greatly limiting applicability. We devise an efficient technique for exact multi-task GP sampling that combines exploiting Kronecker structure in the covariance matrices with Matheron’s identity, allowing us to perform Bayesian optimization using exact multi-task GP models with tens of thousands of correlated outputs. In doing so, we achieve substantial improvements in sample efficiency compared to existing approaches that model solely the outcome metrics. We demonstrate how this unlocks a new class of applications for Bayesian optimization across a range of tasks in science and engineering, including optimizing interference patterns of an optical interferometer with 65,000 outputs.more » « less
-
Bayesian inference provides a systematic framework for integration of data with mathematical models to quantify the uncertainty in the solution of the inverse problem. However, the solution of Bayesian inverse problems governed by complex forward models described by partial differential equations (PDEs) remains prohibitive with black-box Markov chain Monte Carlo (MCMC) methods. We present hIPPYlib-MUQ, an extensible and scalable software framework that contains implementations of state-of-the art algorithms aimed to overcome the challenges of high-dimensional, PDE-constrained Bayesian inverse problems. These algorithms accelerate MCMC sampling by exploiting the geometry and intrinsic low-dimensionality of parameter space via derivative information and low rank approximation. The software integrates two complementary open-source software packages, hIPPYlib and MUQ. hIPPYlib solves PDE-constrained inverse problems using automatically-generated adjoint-based derivatives, but it lacks full Bayesian capabilities. MUQ provides a spectrum of powerful Bayesian inversion models and algorithms, but expects forward models to come equipped with gradients and Hessians to permit large-scale solution. By combining these two complementary libraries, we created a robust, scalable, and efficient software framework that realizes the benefits of each and allows us to tackle complex large-scale Bayesian inverse problems across a broad spectrum of scientific and engineering disciplines. To illustrate the capabilities of hIPPYlib-MUQ, we present a comparison of a number of MCMC methods available in the integrated software on several high-dimensional Bayesian inverse problems. These include problems characterized by both linear and nonlinear PDEs, various noise models, and different parameter dimensions. The results demonstrate that large (∼ 50×) speedups over conventional black box and gradient-based MCMC algorithms can be obtained by exploiting Hessian information (from the log-posterior), underscoring the power of the integrated hIPPYlib-MUQ framework.more » « less
-
null (Ed.)Bayesian hybrid models fuse physics-based insights with machine learning constructs to correct for systematic bias. In this paper, we compare Bayesian hybrid models against physics-based glass-box and Gaussian process black-box surrogate models. We consider ballistic firing as an illustrative case study for a Bayesian decision-making workflow. First, Bayesian calibration is performed to estimate model parameters. We then use the posterior distribution from Bayesian analysis to compute optimal firing conditions to hit a target via a single-stage stochastic program. The case study demonstrates the ability of Bayesian hybrid models to overcome systematic bias from missing physics with fewer data than the pure machine learning approach. Ultimately, we argue Bayesian hybrid models are an emerging paradigm for data-informed decision-making under parametric and epistemic uncertainty.more » « less
-
Robust optimization (RO) is a popular paradigm for modeling and solving two- and multistage decision-making problems affected by uncertainty. In many real-world applications, such as R&D project selection, production planning, or preference elicitation for product or policy recommendations, the time of information discovery is decision-dependent and the uncertain parameters only become observable after an often costly investment. Yet, most of the literature on robust optimization assumes that the uncertain parameters can be observed for free and that the sequence in which they are revealed is independent of the decision-maker’s actions. To fill this gap in the practicability of RO, we consider two- and multistage robust optimization problems in which part of the decision variables control the time of information discovery. Thus, information available at any given time is decision-dependent and can be discovered (at least in part) by making strategic exploratory investments in previous stages. We propose a novel dynamic formulation of the problem and prove its correctness. We leverage our model to provide a solution method inspired from the K-adaptability approximation, whereby K candidate strategies for each decision stage are chosen here-and-now and, at the beginning of each period, the best of these strategies is selected after the uncertain parameters that were chosen to be observed are revealed. We reformulate the problem as a finite mixed-integer (resp. bilinear) program if none (resp. some) of the decision variables are real-valued. This finite program is solvable with off-the-shelf solvers. We generalize our approach to the minimization of piecewise linear convex functions. We demonstrate the effectiveness of our method in terms of usability, optimality, and speed on synthetic instances of the Pandora box problem, the preference elicitation problem with real-valued recommendations, the best box problem, and the R&D project portfolio optimization problem. Finally, we evaluate it on an instance of the active preference elicitation problem used to recommend kidney allocation policies to policy-makers at the United Network for Organ Sharing based on real data from the U.S. Kidney Allocation System. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported primarily by the Operations Engineering Program of the National Science Foundation under NSF Award No. 1763108. The authors are grateful for this support. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2021.00160 .more » « less
