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1. Learning and Optimization with Bayesian Hybrid Models

   https://doi.org/10.23919/ACC45564.2020.9148007  

   Eugene, E. A; Gao, X.; Dowling, A. W. (July 2021, 2020 American Control Conference (ACC))

   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. 

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2. Scalable Stochastic Programming with Bayesian Hybrid Models

   https://doi.org/10.1016/B978-0-323-85159-6.50218-9  

   Wang, J.W.; Eugene, E.A.; Dowling, A.W. (July 2022, Computer aided chemical engineering)

   Yamashita, Y.; Kano, M. (Ed.)

   Bayesian hybrid models (BHMs) fuse physics-based insights with machine learning constructs to correct for systematic bias. In this paper, we demonstrate a scalable computational strategy to embed BHMs in an equation-oriented modelling environment. Thus, this paper generalizes stochastic programming, which traditionally focuses on aleatoric uncertainty (as characterized by a probability distribution for uncertainty model parameters) to also consider epistemic uncertainty, i.e., mode-form uncertainty or systematic bias as modelled by the Gaussian process in the BHM. As an illustrative example, we consider ballistic firing using a BHM that includes a simplified glass-box (i.e., equation-oriented) model that neglects air resistance and a Gaussian process model to account for systematic bias (i.e., epistemic or model-form uncertainty) induced from the model simplification. The gravity parameter and the GP hypermeters are inferred from data in a Bayesian framework, yielding a posterior distribution. A novel single-stage stochastic program formulation using the posterior samples and Gaussian quadrature rules is proposed to compute the optimal decisions (e.g., firing angle and velocity) that minimize the expected value of an objective (e.g., distance from a stationary target). PySMO is used to generate expressions for the GP prediction mean and uncertainty in Pyomo, enabling efficient optimization with gradient-based solvers such as Ipopt. A scaling study characterizes the solver time and number of iterations for up to 2,000 samples from the posterior. 

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3. BONSAI: Structure-exploiting robust Bayesian optimization for networked black-box systems under uncertainty

   https://doi.org/10.1016/j.compchemeng.2025.109393  

   Kudva, Akshay; Paulson, Joel A (January 2026, Computers & Chemical Engineering)

   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. 

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4. Bayesian Optimization with Conformal Prediction Sets

   Stanton, S.; Maddox, W.; Wilson, A.G. (January 2023, Artificial Intelligence and Statistics)

   Bayesian optimization is a coherent, ubiquitous approach to decision-making under uncertainty, with applications including multi-arm bandits, active learning, and black-box optimization. Bayesian optimization selects decisions (i.e. objective function queries) with maximal expected utility with respect to the posterior distribution of a Bayesian model, which quantifies reducible, epistemic uncertainty about query outcomes. In practice, subjectively implausible outcomes can occur regularly for two reasons: 1) model misspecification and 2) covariate shift. Conformal prediction is an uncertainty quantification method with coverage guarantees even for misspecified models and a simple mechanism to correct for covariate shift. We propose conformal Bayesian optimization, which directs queries towards regions of search space where the model predictions have guaranteed validity, and investigate its behavior on a suite of black-box optimization tasks and tabular ranking tasks. In many cases we find that query coverage can be significantly improved without harming sample-efficiency. 

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5. Integrating Hybrid Modeling and Multifidelity Approaches for Data-Driven Process Model Discovery

   https://doi.org/10.69997/sct.151585  

   Ravutla, Suryateja; Boukouvala, Fani (July 2024, Living Archive for Process Systems Engineering)

   Modeling the non-linear dynamics of a system from measurement data accurately is an open challenge. Over the past few years, various tools such as SINDy and DySMHO have emerged as approaches to distill dynamics from data. However, challenges persist in accurately capturing dynamics of a system especially when the physical knowledge about the system is unknown. A promising solution is to use a hybrid paradigm, that combines mechanistic and black-box models to leverage their respective strengths. In this study, we combine a hybrid modeling paradigm with sparse regression, to develop and identify models simultaneously. Two methods are explored, considering varying complexities, data quality, and availability and by comparing different case studies. In the first approach, we integrate SINDy-discovered models with neural ODE structures, to model unknown physics. In the second approach, we employ Multifidelity Surrogate Models (MFSMs) to construct composite models comprised of SINDy-discovered models and error-correction models. 

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