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1. Globally Approximate Gaussian Processes for Big Data With Application to Data-Driven Metamaterials Design

   https://doi.org/10.1115/1.4044257  

   Bostanabad, Ramin; Chan, Yu-Chin; Wang, Liwei; Zhu, Ping; Chen, Wei (November 2019, Journal of Mechanical Design)

   Abstract We introduce a novel method for Gaussian process (GP) modeling of massive datasets called globally approximate Gaussian process (GAGP). Unlike most large-scale supervised learners such as neural networks and trees, GAGP is easy to fit and can interpret the model behavior, making it particularly useful in engineering design with big data. The key idea of GAGP is to build a collection of independent GPs that use the same hyperparameters but randomly distribute the entire training dataset among themselves. This is based on our observation that the GP hyperparameter approximations change negligibly as the size of the training data exceeds a certain level, which can be estimated systematically. For inference, the predictions from all GPs in the collection are pooled, allowing the entire training dataset to be efficiently exploited for prediction. Through analytical examples, we demonstrate that GAGP achieves very high predictive power matching (and in some cases exceeding) that of state-of-the-art supervised learning methods. We illustrate the application of GAGP in engineering design with a problem on data-driven metamaterials, using it to link reduced-dimension geometrical descriptors of unit cells and their properties. Searching for new unit cell designs with desired properties is then achieved by employing GAGP in inverse optimization. 

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2. Adaptive Sampling to Reduce Epistemic Uncertainty Using Prediction Interval-Generation Neural Networks

   https://doi.org/10.1609/aaai.v39i18.34152  

   Morales, Giorgio; Sheppard, John W (April 2025, Proceedings of the AAAI Conference on Artificial Intelligence)

   Obtaining high certainty in predictive models is crucial for making informed and trustworthy decisions in many scientific and engineering domains. However, extensive experimentation required for model accuracy can be both costly and time-consuming. This paper presents an adaptive sampling approach designed to reduce epistemic uncertainty in predictive models. Our primary contribution is the development of a metric that estimates potential epistemic uncertainty leveraging prediction interval-generation neural networks.This estimation relies on the distance between the predicted upper and lower bounds and the observed data at the tested positions and their neighboring points. Our second contribution is the proposal of a batch sampling strategy based on Gaussian processes (GPs). A GP is used as a surrogate model of the networks trained at each iteration of the adaptive sampling process. Using this GP, we design an acquisition function that selects a combination of sampling locations to maximize the reduction of epistemic uncertainty across the domain.We test our approach on three unidimensional synthetic problems and a multi-dimensional dataset based on an agricultural field for selecting experimental fertilizer rates.The results demonstrate that our method consistently converges faster to minimum epistemic uncertainty levels compared to Normalizing Flows Ensembles, MC-Dropout, and simple GPs. 

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3. Scalable Gaussian Processes for Data-Driven Design Using Big Data With Categorical Factors

   https://doi.org/10.1115/1.4052221  

   Wang, Liwei; Yerramilli, Suraj; Iyer, Akshay; Apley, Daniel; Zhu, Ping; Chen, Wei (February 2022, Journal of Mechanical Design)

   null (Ed.)

   Abstract Scientific and engineering problems often require the use of artificial intelligence to aid understanding and the search for promising designs. While Gaussian processes (GP) stand out as easy-to-use and interpretable learners, they have difficulties in accommodating big data sets, categorical inputs, and multiple responses, which has become a common challenge for a growing number of data-driven design applications. In this paper, we propose a GP model that utilizes latent variables and functions obtained through variational inference to address the aforementioned challenges simultaneously. The method is built upon the latent-variable Gaussian process (LVGP) model where categorical factors are mapped into a continuous latent space to enable GP modeling of mixed-variable data sets. By extending variational inference to LVGP models, the large training data set is replaced by a small set of inducing points to address the scalability issue. Output response vectors are represented by a linear combination of independent latent functions, forming a flexible kernel structure to handle multiple responses that might have distinct behaviors. Comparative studies demonstrate that the proposed method scales well for large data sets with over 104 data points, while outperforming state-of-the-art machine learning methods without requiring much hyperparameter tuning. In addition, an interpretable latent space is obtained to draw insights into the effect of categorical factors, such as those associated with “building blocks” of architectures and element choices in metamaterial and materials design. Our approach is demonstrated for machine learning of ternary oxide materials and topology optimization of a multiscale compliant mechanism with aperiodic microstructures and multiple materials. 

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4. Conditional Deep Gaussian Processes: Empirical Bayes Hyperdata Learning

   https://doi.org/10.3390/e23111387  

   Lu, Chi-Ken; Shafto, Patrick (November 2021, Entropy)

   null (Ed.)

   It is desirable to combine the expressive power of deep learning with Gaussian Process (GP) in one expressive Bayesian learning model. Deep kernel learning showed success as a deep network used for feature extraction. Then, a GP was used as the function model. Recently, it was suggested that, albeit training with marginal likelihood, the deterministic nature of a feature extractor might lead to overfitting, and replacement with a Bayesian network seemed to cure it. Here, we propose the conditional deep Gaussian process (DGP) in which the intermediate GPs in hierarchical composition are supported by the hyperdata and the exposed GP remains zero mean. Motivated by the inducing points in sparse GP, the hyperdata also play the role of function supports, but are hyperparameters rather than random variables. It follows our previous moment matching approach to approximate the marginal prior for conditional DGP with a GP carrying an effective kernel. Thus, as in empirical Bayes, the hyperdata are learned by optimizing the approximate marginal likelihood which implicitly depends on the hyperdata via the kernel. We show the equivalence with the deep kernel learning in the limit of dense hyperdata in latent space. However, the conditional DGP and the corresponding approximate inference enjoy the benefit of being more Bayesian than deep kernel learning. Preliminary extrapolation results demonstrate expressive power from the depth of hierarchy by exploiting the exact covariance and hyperdata learning, in comparison with GP kernel composition, DGP variational inference and deep kernel learning. We also address the non-Gaussian aspect of our model as well as way of upgrading to a full Bayes inference. 

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5. Surrogate modeling for Bayesian optimization beyond a single Gaussian process

   https://doi.org/10.1109/TPAMI.2023.3264741  

   Lu, Qin; Polyzos, Konstantinos D.; Li, Bingcong; Giannakis, Georgios B. (September 2023, IEEE Transactions on Pattern Analysis and Machine Intelligence)

   Bayesian optimization (BO) has well-documented merits for optimizing black-box functions with an expensive evaluation cost. Such functions emerge in applications as diverse as hyperparameter tuning, drug discovery, and robotics. BO hinges on a Bayesian surrogate model to sequentially select query points so as to balance exploration with exploitation of the search space. Most existing works rely on a single Gaussian process (GP) based surrogate model, where the kernel function form is typically preselected using domain knowledge. To bypass such a design process, this paper leverages an ensemble (E) of GPs to adaptively select the surrogate model fit on-the-fly, yielding a GP mixture posterior with enhanced expressiveness for the sought function. Acquisition of the next evaluation input using this EGP-based function posterior is then enabled by Thompson sampling (TS) that requires no additional design parameters. To endow function sampling with scalability, random feature-based kernel approximation is leveraged per GP model. The novel EGP-TS readily accommodates parallel operation. To further establish convergence of the proposed EGP-TS to the global optimum, analysis is conducted based on the notion of Bayesian regret for both sequential and parallel settings. Tests on synthetic functions and real-world applications showcase the merits of the proposed method. 

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