Molecular property prediction (e.g., energy) is an essential problem in chemistry and biology. Unfortunately, many supervised learning methods usually suffer from the problem of scarce labeled molecules in the chemical space, where such property labels are generally obtained by Density Functional Theory (DFT) calculation which is extremely computational costly. An effective solution is to incorporate the unlabeled molecules in a semi-supervised fashion. However, learning semi-supervised representation for large amounts of molecules is challenging, including the joint representation issue of both molecular essence and structure, the conflict between representation and property leaning. Here we propose a novel framework called Active Semi-supervised Graph Neural Network (ASGN) by incorporating both labeled and unlabeled molecules. Specifically, ASGN adopts a teacher-student framework. In the teacher model, we propose a novel semi-supervised learning method to learn general representation that jointly exploits information from molecular structure and molecular distribution. Then in the student model, we target at property prediction task to deal with the learning loss conflict. At last, we proposed a novel active learning strategy in terms of molecular diversities to select informative data during the whole framework learning. We conduct extensive experiments on several public datasets. Experimental results show the remarkable performance of our ASGN framework.

Physics-informed neural networks (PINNs) have become a popular choice for solving high-dimensional partial differential equations (PDEs) due to their excellent approximation power and generalization ability. Recently, Extended PINNs (XPINNs) based on domain decomposition methods have attracted considerable attention due to their effectiveness in modeling multiscale and multiphysics problems and their parallelization. However, theoretical understanding on their convergence and generalization properties remains unexplored. In this study, we take an initial step towards understanding how and when XPINNs outperform PINNs. Specifically, for general multi-layer PINNs and XPINNs, we first provide a prior generalization bound via the complexity of the target functions in the PDE problem, and a posterior generalization bound via the posterior matrix norms of the networks after optimization. Moreover, based on our bounds, we analyze the conditions under which XPINNs improve generalization. Concretely, our theory shows that the key building block of XPINN, namely the domain decomposition, introduces a tradeoff for generalization. On the one hand, XPINNs decompose the complex PDE solution into several simple parts, which decreases the complexity needed to learn each part and boosts generalization. On the other hand, decomposition leads to less training data being available in each subdomain, and hence such model is typically prone to overfitting and may become less generalizable. Empirically, we choose five PDEs to show when XPINNs perform better than, similar to, or worse than PINNs, hence demonstrating and justifying our new theory.

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Z. (2020) 'A comparative analysis of location and non-location-bounded strategic asset seeking in emerging and developed market MNEs : an application of new internalization theory.', International business review., 29 (2). p. 101635.

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