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Graph neural networks (GNNs) have been applied to a large variety of applications in materials science and chemistry. Here, we recapitulate the graph construction for crystalline (periodic) materials and investigate its impact on the GNNs model performance. We suggest the asymmetric unit cell as a representation to reduce the number of atoms by using all symmetries of the system. This substantially reduced the computational cost and thus time needed to train large graph neural networks without any loss in accuracy. Furthermore, with a simple but systematically built GNN architecture based on message passing and line graph templates, we introduce a general architecture (Nested Graph Network, NGN) that is applicable to a wide range of tasks. We show that our suggested models systematically improve state-of-the-art results across all tasks within the MatBench benchmark. Further analysis shows that optimized connectivity and deeper message functions are responsible for the improvement. Asymmetric unit cells and connectivity optimization can be generally applied to (crystal) graph networks, while our suggested nested graph framework will open new ways of systematic comparison of GNN architectures.

Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates efficient numerical methods. In this paper, we propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet). Our proposed method leverages the power of deep learning in conjunction with traditional numerical methods, specifically the finite element method, to solve parametric PDEs in the absence of any paired input-output training data. We demonstrate the effectiveness of our approach on several benchmark problems and show that it outperforms existing state-of-the-art methods in terms of accuracy, generalization, and computational flexibility. Our FEONet framework shows potential for application in various fields where PDEs play a crucial role in modeling complex domains with diverse boundary conditions and singular behavior. Furthermore, we provide theoretical convergence analysis to support our approach, utilizing finite element approximation in numerical analysis.

Extracting level sets from scalar data is a fundamental operation in visualization with many applications. Recently, the concept of level set extraction has been extended to bivariate scalar fields. Prior work on vector field equivalence, wherein an analyst marks a region in the domain and is shown other regions in the domain with similar vector values, pointed out the need to make this extraction operation fast, so that analysts can work interactively. To date, the fast extraction of level sets from bivariate scalar fields has not been researched as extensively as for the univariate case. In this paper, we present a novel algorithm that extracts fiber lines, i.e., the preimages of so called control polygons (FSCP), for bivariate 2D data by joint traversal of bounding volume hierarchies for both grid and FSCP elements. We performed an extensive evaluation, comparing our method to a two-dimensional adaptation of the method proposed by Klacansky et al., as well as to the naive approach for fiber line extraction. The evaluation incorporates a vast array of configurations in several datasets. We found that our method provides a speedup of several orders of magnitudes compared to the naive algorithm and requires two thirds of the computation time compared to Klacansky et al. adapted for 2D.

Interface problems have long been a major focus of scientific computing, leading to the development of various numerical methods. Traditional mesh-based methods often employ time-consuming body-fitted meshes with standard discretization schemes or unfitted meshes with tailored schemes to achieve controllable accuracy and convergence rate. Along another line, mesh-free methods bypass mesh generation but lack robustness in terms of convergence and accuracy due to the low regularity of solutions. In this study, we propose a novel method for solving interface problems within the framework of the random feature method. This approach utilizes random feature functions in conjunction with a partition of unity as approximation functions. It evaluates partial differential equations, boundary conditions, and interface conditions on collocation points in equal footing, and solves a linear least-squares system to obtain the approximate solution. To address the issue of low regularity, two sets of random feature functions are used to approximate the solution on each side of the interface, which are then coupled together via interface conditions. We validate our method through a series of increasingly complex numerical examples. Our findings show that despite the solution often being only continuous or even discontinuous, our method not only eliminates the need for mesh generation but also maintains high accuracy, akin to the spectral collocation method for smooth solutions. Remarkably, for the same accuracy requirement, our method requires two to three orders of magnitude fewer degrees of freedom than traditional methods, demonstrating its significant potential for solving interface problems with complex geometries.

There is rising interest in differentiable rendering, which allows explicitly modeling geometric priors and constraints in optimization pipelines using first-order methods such as backpropagation. Incorporating such domain knowledge can lead to deep neural networks that are trained more robustly and with limited data, as well as the capability to solve ill-posed inverse problems. Existing efforts in differentiable rendering have focused on imagery from electro-optical sensors, particularly conventional RGB-imagery. In this work, we propose an approach for differentiable rendering of Synthetic Aperture Radar (SAR) imagery, which combines methods from 3D computer graphics with neural rendering. We demonstrate the approach on the inverse graphics problem of 3D Object Reconstruction from limited SAR imagery using high-fidelity simulated SAR data.

Data management has traditionally relied on synthetic data generators to generate structured benchmarks, like the TPC suite, where we can control important parameters like data size and its distribution precisely. These benchmarks were central to the success and adoption of database management systems. But more and more, data management problems are of a semantic nature. An important example is finding tables that can be unioned. While any two tables with the same cardinality can be unioned, table union search is the problem of finding tables whose union is semantically coherent. Semantic problems cannot be benchmarked using synthetic data. Our current methods for creating benchmarks involve the manual curation and labeling of real data. These methods are not robust or scalable and perhaps more importantly, it is not clear how robust the created benchmarks are. We propose to use generative AI models to create structured data benchmarks for table union search. We present a novel method for using generative models to create tables with specified properties. Using this method, we create a new benchmark containing pairs of tables that are both unionable and non-unionable but related. We thoroughly evaluate recent existing table union search methods over existing benchmarks and our new benchmark. We also present and evaluate a new table search methods based on recent large language models over all benchmarks. We show that the new benchmark is more challenging for all methods than hand-curated benchmarks, specifically, the top-performing method achieves a Mean Average Precision of around 60%, over 30% less than its performance on existing manually created benchmarks. We examine why this is the case and show that the new benchmark permits more detailed analysis of methods, including a study of both false positives and false negatives that were not possible with existing benchmarks.

Data augmentation has been widely used to improve generalizability of machine learning models. However, comparatively little work studies data augmentation for graphs. This is largely due to the complex, non-Euclidean structure of graphs, which limits possible manipulation operations. Augmentation operations commonly used in vision and language have no analogs for graphs. Our work studies graph data augmentation for graph neural networks (GNNs) in the context of improving semi-supervised node-classification. We discuss practical and theoretical motivations, considerations and strategies for graph data augmentation. Our work shows that neural edge predictors can effectively encode class-homophilic structure to promote intra-class edges and demote inter-class edges in given graph structure, and our main contribution introduces the GAug graph data augmentation framework, which leverages these insights to improve performance in GNN-based node classification via edge prediction. Extensive experiments on multiple benchmarks show that augmentation via GAug improves performance across GNN architectures and datasets.

Deep learning methods for graphs achieve remarkable performance on many node-level and graph-level prediction tasks. However, despite the proliferation of the methods and their success, prevailing Graph Neural Networks (GNNs) neglect subgraphs, rendering subgraph prediction tasks challenging to tackle in many impactful applications. Further, subgraph prediction tasks present several unique challenges, because subgraphs can have non-trivial internal topology, but also carry a notion of position and external connectivity information relative to the underlying graph in which they exist. Here, we introduce SUB-GNN, a subgraph neural network to learn disentangled subgraph representations. In particular, we propose a novel subgraph routing mechanism that propagates neural messages between the subgraph's components and randomly sampled anchor patches from the underlying graph, yielding highly accurate subgraph representations. SUB-GNN specifies three channels, each designed to capture a distinct aspect of subgraph structure, and we provide empirical evidence that the channels encode their intended properties. We design a series of new synthetic and real-world subgraph datasets. Empirical results for subgraph classification on eight datasets show that SUB-GNN achieves considerable performance gains, outperforming strong baseline methods, including node-level and graph-level GNNs, by 12.4% over the strongest baseline. SUB-GNN performs exceptionally well on challenging biomedical datasets when subgraphs have complex topology and even comprise multiple disconnected components.

Knowledge graph embedding, which aims to represent entities and relations as low dimensional vectors (or matrices, tensors, etc.), has been shown to be a powerful technique for predicting missing links in knowledge graphs. Existing knowledge graph embedding models mainly focus on modeling relation patterns such as symmetry/antisymmetry, inversion, and composition. However, many existing approaches fail to model semantic hierarchies, which are common in real-world applications. To address this challenge, we propose a novel knowledge graph embedding model---namely, Hierarchy-Aware Knowledge Graph Embedding (HAKE)---which maps entities into the polar coordinate system. HAKE is inspired by the fact that concentric circles in the polar coordinate system can naturally reflect the hierarchy. Specifically, the radial coordinate aims to model entities at different levels of the hierarchy, and entities with smaller radii are expected to be at higher levels; the angular coordinate aims to distinguish entities at the same level of the hierarchy, and these entities are expected to have roughly the same radii but different angles. Experiments demonstrate that HAKE can effectively model the semantic hierarchies in knowledge graphs, and significantly outperforms existing state-of-the-art methods on benchmark datasets for the link prediction task.

The potential of graph convolutional neural networks for the task of zero-shot learning has been demonstrated recently. These models are highly sample efficient as related concepts in the graph structure share statistical strength allowing generalization to new classes when faced with a lack of data. However, knowledge from distant nodes can get diluted when propagating through intermediate nodes, because current approaches to zero-shot learning use graph propagation schemes that perform Laplacian smoothing at each layer. We show that extensive smoothing does not help the task of regressing classifier weights in zero-shot learning. In order to still incorporate information from distant nodes and utilize the graph structure, we propose an Attentive Dense Graph Propagation Module (ADGPM). ADGPM allows us to exploit the hierarchical graph structure of the knowledge graph through additional connections. These connections are added based on a node's relationship to its ancestors and descendants and an attention scheme is further used to weigh their contribution depending on the distance to the node. Finally, we illustrate that finetuning of the feature representation after training the ADGPM leads to considerable improvements. Our method achieves competitive results, outperforming previous zero-shot learning approaches.