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Deep learning has been a prevalence in computational chemistry and widely implemented in molecule property predictions. Recently, self-supervised learning (SSL), especially contrastive learning (CL), gathers growing attention for the potential to learn molecular representations that generalize to the gigantic chemical space. Unlike supervised learning, SSL can directly leverage large unlabeled data, which greatly reduces the effort to acquire molecular property labels through costly and time-consuming simulations or experiments. However, most molecular SSL methods borrow the insights from the machine learning community but neglect the unique cheminformatics (e.g., molecular fingerprints) and multi-level graphical structures (e.g., functional groups) of molecules. In this work, we propose iMolCLR: improvement of Molecular Contrastive Learning of Representations with graph neural networks (GNNs) in two aspects, (1) mitigating faulty negative contrastive instances via considering cheminformatics similarities between molecule pairs; (2) fragment-level contrasting between intra- and inter-molecule substructures decomposed from molecules. Experiments have shown that the proposed strategies significantly improve the performance of GNN models on various challenging molecular property predictions. In comparison to the previous CL framework, iMolCLR demonstrates an averaged 1.3% improvement of ROC-AUC on 7 classification benchmarks and an averaged 4.8% decrease of the error on 5 regression benchmarks. On most benchmarks, the generic GNN pre-trained by iMolCLR rivals or even surpasses supervised learning models with sophisticated architecture designs and engineered features. Further investigations demonstrate that representations learned through iMolCLR intrinsically embed scaffolds and functional groups that can reason molecule similarities.