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Supervised machine learning approaches have been increasingly used in accelerating electronic structure prediction as surrogates of first-principle computational methods, such as density functional theory (DFT). While numerous quantum chemistry datasets focus on chemical properties and atomic forces, the ability to achieve accurate and efficient prediction of the Hamiltonian matrix is highly desired, as it is the most important and fundamental physical quantity that determines the quantum states of physical systems and chemical properties. In this work, we generate a new Quantum Hamiltonian dataset, named as QH9, to provide precise Hamiltonian matrices for 2,399 molecular dynamics trajectories and 130,831 stable molecular geometries, based on the QM9 dataset. By designing benchmark tasks with various molecules, we show that current machine learning models have the capacity to predict Hamiltonian matrices for arbitrary molecules. Both the QH9 dataset and the baseline models are provided to the community through an open-source benchmark, which can be highly valuable for developing machine learning methods and accelerating molecular and materials design for scientific and technological applications. Our benchmark is publicly available at \url{https://github.com/divelab/AIRS/tree/main/OpenDFT/QHBench}. 

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. 

We consider the prediction of the Hamiltonian matrix, which finds use in quantum chemistry and condensed matter physics. Efficiency and equivariance are two important, but conflicting factors. In this work, we propose a SE(3)-equivariant network, named QHNet, that achieves efficiency and equivariance. Our key advance lies at the innovative design of QHNet architecture, which not only obeys the underlying symmetries, but also enables the reduction of number of tensor products by 92%. In addition, QHNet prevents the exponential growth of channel dimension when more atom types are involved. We perform experiments on MD17 datasets, including four molecular systems. Experimental results show that our QHNet can achieve comparable performance to the state of the art methods at a significantly faster speed. Besides, our QHNet consumes 50% less memory due to its streamlined architecture. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS). 

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.