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Thanks to the rapid advances in artificial intelligence, AI for science (AI4Science) has emerged as one of the new promising research directions for modern science and engineering. In this review, we focus on recent efforts to develop knowledge-driven Bayesian learning and experimental design methods for accelerating the discovery of novel functional materials as well as enhancing the understanding of composition-process-structure-property relationships. We specifically discuss the challenges and opportunities in integrating prior scientific knowledge and physics principles with AI and machine learning (ML) models for accelerating materials and knowledge discovery. The current state-of-the-art methods in knowledge-based prior construction, model fusion, uncertainty quantification, optimal experimental design, and symbolic regression are detailed in the review, along with several detailed case studies and results in materials discovery. 

Designing and/or controlling complex systems in science and engineering relies on appropriate mathematical modeling of systems dynamics. Classical differential equation based solutions in applied and computational mathematics are often computationally demanding. Recently, the connection between reduced-order models of high-dimensional differential equation systems and surrogate machine learning models has been explored. However, the focus of both existing reduced-order and machine learning models for complex systems has been how to best approximate the high fidelity model of choice. Due to high complexity and often limited training data to derive reduced-order or machine learning surrogate models, it is critical for derived reduced-order models to have reliable uncertainty quantification at the same time. In this paper, we propose such a novel framework of Bayesian reduced-order models naturally equipped with uncertainty quantification as it learns the distributions of the parameters of the reduced-order models instead of their point estimates. In particular, we develop learnable Bayesian proper orthogonal decomposition (BayPOD) that learns the distributions of both the POD projection bases and the mapping from the system input parameters to the projected scores/coefficients so that the learned BayPOD can help predict high-dimensional systems dynamics/fields as quantities of interest in different setups with reliable uncertainty estimates. The developed learnable BayPOD inherits the capability of embedding physics constraints when learning the POD-based surrogate reduced-order models, a desirable feature when studying complex systems in science and engineering applications where the available training data are limited. Furthermore, the proposed BayPOD method is an end-to-end solution, which unlike other surrogate-based methods, does not require separate POD and machine learning steps. The results from a real-world case study of the pressure field around an airfoil.