Geometric deep learning (GDL) has demonstrated huge power and enormous potential in molecular data analysis. However, a great challenge still remains for highly efficient molecular representations. Currently, covalent-bond-based molecular graphs are the de facto standard for representing molecular topology at the atomic level. Here we demonstrate, for the first time, that molecular graphs constructed only from non-covalent bonds can achieve similar or even better results than covalent-bond-based models in molecular property prediction. This demonstrates the great potential of novel molecular representations beyond the de facto standard of covalent-bond-based molecular graphs. Based on the finding, we propose molecular geometric deep learning (Mol-GDL). The essential idea is to incorporate a more general molecular representation into GDL models. In our Mol-GDL, molecular topology is modeled as a series of molecular graphs, each focusing on a different scale of atomic interactions. In this way, both covalent interactions and non-covalent interactions are incorporated into the molecular representation on an equal footing. We systematically test Mol-GDL on fourteen commonly-used benchmark datasets. The results show that our Mol-GDL can achieve a better performance than state-of-the-art (SOTA) methods. Source code and data are available at https://github.com/CS-BIO/Mol-GDL.
Geometric deep learning has demonstrated a great potential in non-Euclidean data analysis. The incorporation of geometric insights into learning architecture is vital to its success. Here we propose a curvature-enhanced graph convolutional network (CGCN) for biomolecular interaction prediction, for the first time. Our CGCN employs Ollivier-Ricci curvature (ORC) to characterize network local structures and to enhance the learning capability of GCNs. More specifically, ORCs are evaluated based on the local topology from node neighborhoods, and further used as weights for the feature aggregation in message-passing procedure. Our CGCN model is extensively validated on fourteen real-world bimolecular interaction networks and a series of simulated data. It has been found that our CGCN can achieve the state-of-the-art results. It outperforms all existing models, as far as we know, in thirteen out of the fourteen real-world datasets and ranks as the second in the rest one. The results from the simulated data show that our CGCN model is superior to the traditional GCN models regardless of the positive-to-negativecurvature ratios, network densities, and network sizes (when larger than 500).
Traditionally, graph neural networks have been trained using a single observed graph. However, the observed graph represents only one possible realization. In many applications, the graph may encounter uncertainties, such as having erroneous or missing edges, as well as edge weights that provide little informative value. To address these challenges and capture additional information previously absent in the observed graph, we introduce latent variables to parameterize and generate multiple graphs. We obtain the maximum likelihood estimate of the network parameters in an Expectation-Maximization (EM) framework based on the multiple graphs. Specifically, we iteratively determine the distribution of the graphs using a Markov Chain Monte Carlo (MCMC) method, incorporating the principles of PAC-Bayesian theory. Numerical experiments demonstrate improvements in performance against baseline models on node classification for heterogeneous graphs and graph regression on chemistry datasets.
Geometric deep learning (GDL) models have demonstrated a great potential for the analysis of non-Euclidian data. They are developed to incorporate the geometric and topological information of non-Euclidian data into the end-to-end deep learning architectures. Motivated by the recent success of discrete Ricci curvature in graph neural network (GNNs), we propose TorGNN, an analytic Torsion enhanced Graph Neural Network model. The essential idea is to characterize graph local structures with an analytic torsion based weight formula. Mathematically, analytic torsion is a topological invariant that can distinguish spaces which are homotopy equivalent but not homeomorphic. In our TorGNN, for each edge, a corresponding local simplicial complex is identified, then the analytic torsion (for this local simplicial complex) is calculated, and further used as a weight (for this edge) in message-passing process. Our TorGNN model is validated on link prediction tasks from sixteen different types of networks and node classification tasks from three types of networks. It has been found that our TorGNN can achieve superior performance on both tasks, and outperform various state-of-the-art models. This demonstrates that analytic torsion is a highly efficient topological invariant in the characterization of graph structures and can significantly boost the performance of GNNs.
Recently, therapeutic peptides have demonstrated great promise for cancer treatment. To explore powerful anticancer peptides, artificial intelligence (AI)-based approaches have been developed to systematically screen potential candidates. However, the lack of efficient featurization of peptides has become a bottleneck for these machine-learning models. In this paper, we propose a topology-enhanced machine learning model (Top-ML) for anticancer peptides prediction. Our Top-ML employs peptide topological features derived from its sequence "connection" information characterized by vector and spectral descriptors. Our Top-ML model, employing an Extra-Trees classifier, has been validated on the AntiCP 2.0 and mACPpred 2.0 benchmark datasets, achieving state-of-the-art performance or results comparable to existing deep learning models, while providing greater interpretability. Our results highlight the potential of leveraging novel topology-based featurization to accelerate the identification of anticancer peptides.
As key models in geometric deep learning, graph neural networks have demonstrated enormous power in molecular data analysis. Recently, a specially-designed learning scheme, known as Kolmogorov-Arnold Network (KAN), shows unique potential for the improvement of model accuracy, efficiency, and explainability. Here we propose the first non-trivial Kolmogorov-Arnold Network-based Graph Neural Networks (KA-GNNs), including KAN-based graph convolutional networks(KA-GCN) and KAN-based graph attention network (KA-GAT). The essential idea is to utilizes KAN's unique power to optimize GNN architectures at three major levels, including node embedding, message passing, and readout. Further, with the strong approximation capability of Fourier series, we develop Fourier series-based KAN model and provide a rigorous mathematical prove of the robust approximation capability of this Fourier KAN architecture. To validate our KA-GNNs, we consider seven most-widely-used benchmark datasets for molecular property prediction and extensively compare with existing state-of-the-art models. It has been found that our KA-GNNs can outperform traditional GNN models. More importantly, our Fourier KAN module can not only increase the model accuracy but also reduce the computational time. This work not only highlights the great power of KA-GNNs in molecular property prediction but also provides a novel geometric deep learning framework for the general non-Euclidean data analysis.
Graph Neural Networks (GNNs) have proven effective for learning from graph-structured data through their neighborhood-based message passing framework. Many hierarchical graph clustering pooling methods modify this framework by introducing clustering-based strategies, enabling the construction of more expressive and powerful models. However, all of these message passing framework heavily rely on the connectivity structure of graphs, limiting their ability to capture the rich geometric features inherent in geometric graphs. To address this, we propose Rhomboid Tiling (RT) clustering, a novel clustering method based on the rhomboid tiling structure, which performs clustering by leveraging the complex geometric information of the data and effectively extracts its higher-order geometric structures. Moreover, we design RTPool, a hierarchical graph clustering pooling model based on RT clustering for graph classification tasks. The proposed model demonstrates superior performance, outperforming 21 state-of-the-art competitors on all the 7 benchmark datasets.
In this paper, we propose persistent spectral based machine learning (PerSpect ML) models for drug design. Persistent spectral models, including persistent spectral graph, persistent spectral simplicial complex and persistent spectral hypergraph, are proposed based on spectral graph theory, spectral simplicial complex theory and spectral hypergraph theory, respectively. Different from all previous spectral models, a filtration process, as proposed in persistent homology, is introduced to generate multiscale spectral models. More specifically, from the filtration process, a series of nested topological representations, i,e., graphs, simplicial complexes, and hypergraphs, can be systematically generated and their spectral information can be obtained. Persistent spectral variables are defined as the function of spectral variables over the filtration value. Mathematically, persistent multiplicity (of zero eigenvalues) is exactly the persistent Betti number (or Betti curve). We consider 11 persistent spectral variables and use them as the feature for machine learning models in protein-ligand binding affinity prediction. We systematically test our models on three most commonly-used databases, including PDBbind-2007, PDBbind-2013 and PDBbind-2016. Our results, for all these databases, are better than all existing models, as far as we know. This demonstrates the great power of our PerSpect ML in molecular data analysis and drug design.
The matched interface and boundary (MIB) method has a proven ability for delivering the second order accuracy in handling elliptic interface problems with arbitrarily complex interface geometries. However, its collocation formulation requires relatively high solution regularity. Finite volume method (FVM) has its merit in dealing with conservation law problems and its integral formulation works well with relatively low solution regularity. We propose an MIB-FVM to take the advantages of both MIB and FVM for solving elliptic interface problems. We construct the proposed method on Cartesian meshes with vertex-centered control volumes. A large number of numerical experiments are designed to validate the present method in both two dimensional (2D) and three dimensional (3D) domains. It is found that the proposed MIB-FVM achieves the second order convergence for elliptic interface problems with complex interface geometries in both $L_{\infty}$ and $L_2$ norms.
Elasticity theory is an important component of continuum mechanics and has had widely spread applications in science and engineering. Material interfaces are ubiquity in nature and man-made devices, and often give rise to discontinuous coefficients in the governing elasticity equations. In this work, the matched interface and boundary (MIB) method is developed to address elasticity interface problems. Linear elasticity theory for both isotropic homogeneous and inhomogeneous media is employed. In our approach, Lam$\acute{e}$'s parameters can have jumps across the interface and are allowed to be position dependent in modeling isotropic inhomogeneous material. Both strong discontinuity, i.e., discontinuous solution, and weak discontinuity, namely, discontinuous derivatives of the solution, are considered in the present study. In the proposed method, fictitious values are utilized so that the standard central finite different schemes can be employed regardless of the interface. Interface jump conditions are enforced on the interface, which in turn, accurately determines fictitious values. We design new MIB schemes to account for complex interface geometries. In particular, the cross derivatives in the elasticity equations are difficult to handle for complex interface geometries. We propose secondary fictitious values and construct geometry based interpolation schemes to overcome this difficulty. Numerous analytical examples are used to validate the accuracy, convergence and robustness of the present MIB method for elasticity interface problems with both small and large curvatures, strong and weak discontinuities, and constant and variable coefficients. Numerical tests indicate second order accuracy in both $L_\infty$ and $L_2$ norms.