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By querying approximate surrogate models of different fidelity as available information sources, Multi-Fidelity Bayesian Optimization (MFBO) aims at optimizing unknown functions that are costly or infeasible to evaluate. Existing MFBO methods often assume that approximate surrogates have consistently high or low fidelity across the input domain. However, approximate evaluations from the same surrogate can have different fidelity at different input regions due to data availability and model constraints, especially when considering machine learning surrogates. In this work, we investigate MFBO when multi-fidelity approximations have input-dependent fidelity. By explicitly capturing input dependency for multi-fidelity queries in a Gaussian Process (GP), our new input-dependent MFBO (iMFBO) with learnable noise models better captures the fidelity of each information source in an intuitive way. We further design a new acquisition function for iMFBO and prove that the queries selected by iMFBO have higher quality than those by naive MFBO methods, with a derived sub-linear regret bound. Experiments on both synthetic and real-world data demonstrate its superior empirical performance. 

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.