In this paper, we propose DGCL, a dual-graph neural networks (GNNs)-based contrastive learning (CL) integrated with mixed molecular fingerprints (MFPs) for molecular property prediction. The DGCL-MFP method contains two stages. In the first pretraining stage, we utilize two different GNNs as encoders to construct CL, rather than using the method of generating enhanced graphs as before. Precisely, DGCL aggregates and enhances features of the same molecule by the Graph Isomorphism Network and the Graph Attention Network, with representations extracted from the same molecule serving as positive samples, and others marked as negative ones. In the downstream tasks training stage, features extracted from the two above pretrained graph networks and the meticulously selected MFPs are concated together to predict molecular properties. Our experiments show that DGCL enhances the performance of existing GNNs by achieving or surpassing the state-of-the-art self-supervised learning models on multiple benchmark datasets. Specifically, DGCL increases the average performance of classification tasks by 3.73$\%$ and improves the performance of regression task Lipo by 0.126. Through ablation studies, we validate the impact of network fusion strategies and MFPs on model performance. In addition, DGCL’s predictive performance is further enhanced by weighting different molecular features based on the Extended Connectivity Fingerprint. The code and datasets of DGCL will be made publicly available.

In this article, we construct a $C^0$ linear finite element method for two fourth-order eigenvalue problems: the biharmonic and the transmission eigenvalue problems. The basic idea of our construction is to use gradient recovery operator to compute the higher-order derivatives of a $C^0$ piecewise linear function, which do not exist in the classical sense. For the biharmonic eigenvalue problem, the optimal convergence rates of eigenvalue/eigenfunction approximation are theoretically derived and numerically verified. For the transmission eigenvalue problem, the optimal convergence rate of the eigenvalues is verified by two numerical examples: one for constant refraction index and the other for variable refraction index. Compared with existing schemes in the literature, the proposed scheme is straightforward and simpler, and computationally less expensive to achieve the same order of accuracy.