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The U.S. Department of Energy (DOE) Office of Integrated Waste Management is planning for the eventual transportation, storage, and disposal of spent nuclear fuel (SNF) and high-level radioactive waste (HLW) from nuclear power plant and DOE sites. The Stakeholder Tool for Assessing Radioactive Transportation (START) is a web-based, geospatial decision-support tool developed for evaluating routing options and other aspects of transporting SNF and HLW, covering rail, truck, barge, and intermodal infrastructure and operations in the continental United States. The verification and validation (V&V) process is intended to independently assess START to provide confidence in the ability of START to accurately provide intended results. The V&V process checks the START tool using a variety of methods, ranging from independent hand calculations to comparison of START performance and results to those of other codes. The V&V activity was conducted independently from the START development team with opportunities to provide feedback and collaborate throughout the process. The V&V analyzed attributes of transportation routes produced by START, including route distance and both population and population density captured within buffer zones around routes. Population in the buffer zone, population density in the buffer zone, and route distance were all identified as crucial outputs of the START code and were subject to V&V tasks. Some of the improvements identified through the V&V process were standardizing the underlying population data in START, changing the projection of the population raster data, and changes to the methodology used for population density to improve its applicability for expected users. This collaboration also led to suggested improvements to some of the underlying shape file segments within START.

Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modelling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT), when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on the choice of coarse scheme. Next, we perform numerical simulations of two standard tests to evaluate the stability, convergence and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight of this code supporting 2-designs. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined. The parameters of the related $2$-designs are also given. Moreover, we prove that the codewords with minimum Hamming weight supporting 3-designs, which gives an affirmative solution to Heng's conjecture.

In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is $C^2$ or polygonal, and the exact solution $(\sigma,u)$ belongs to $H^s(div; \Omega_0 \cup \Omega_1) \times $H^{1+s}(\Omega_0 \cup \Omega_1)$ with $s > 1/2$. Two types of least squares functionals are defined to seek the numerical solution. The first is defined by simply applying the $L^2$ norm least squares principle, and requires the condition $s \geq 1$. The second is defined with a discrete minus norm, which is related to the inner product in $H^{-1/2}(\Gamma)$. The use of this discrete minus norm results in a method of optimal convergence rates and allows the exact solution has the regularity of any $s > 1/2$. The stability near the interface for both methods is guaranteed by the ghost penalty bilinear forms and we can derive the robust condition number estimates. The convergence rates under $L^2$ norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions.

Given a set $P$ of $n$ points in $\mathbb{R}^2$ and an input line $\gamma$ in $\mathbb{R}^2$, we present an algorithm that runs in optimal $\Theta(n\log n)$ time and $\Theta(n)$ space to solve a restricted version of the $1$-Steiner tree problem. Our algorithm returns a minimum-weight tree interconnecting $P$ using at most one Steiner point $s \in \gamma$, where edges are weighted by the Euclidean distance between their endpoints. We then extend the result to $j$ input lines. Following this, we show how the algorithm of Brazil et al. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) that solves the $k$-Steiner tree problem in $\mathbb{R}^2$ in $O(n^{2k})$ time can be adapted to our setting. For $k>1$, restricting the (at most) $k$ Steiner points to lie on an input line, the runtime becomes $O(n^{k})$. Next we show how the results of Brazil et al. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) allow us to maintain the same time and space bounds while extending to some non-Euclidean norms and different tree cost functions. Lastly, we extend the result to $j$ input curves.

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves adjacency, its direction, and its type. The minimum value of $|V(H)|$ such that $G$ admits a homomorphism to $H$ is the $(n,m)$-chromatic number of $G$, denoted by $\mychi_{n,m}(G)$. This parameter was introduced by Ne\v{s}et\v{r}il and Raspaud (J. Comb. Theory. Ser. B 2000). In this article, we show that the arboricity of $G$ is bounded by a function of $\mychi_{n,m}(G)$, but not the other way round. We also show that acyclic chromatic number of $G$ is bounded by a function of $\mychi_{n,m}(G)$, while the other way round bound was known beforehand. Moreover, we show that $(n,m)$-chromatic number for the family of graphs having maximum average degree less than $2+ \frac{2}{4(2n+m)-1}$, which contains the family of planar graphs having girth at least $8(2n+m)$ as a subfamily, is equal to $2(2n+m)+1$. This improves the previously known result which proved that the $(n,m)$-chromatic number for the family planar graphs having girth at least $10(2n+m)-4$ is equal to $2(2n+m)+1$. It is known that the $(n,m)$-chromatic number for the family of partial $2$-trees bounded below and above by quadratic functions of $(2n+m)$ and that the lower bound is tight when $(2n+m)=2$. We show that the lower bound is not tight when $(2n+m)=3$ by improving the corresponding lower bounds by one. We manage to improve some of the upper bounds in these cases as well.

Graph neural networks generalize conventional neural networks to graph-structured data and have received widespread attention due to their impressive representation ability. In spite of the remarkable achievements, the performance of Euclidean models in graph-related learning is still bounded and limited by the representation ability of Euclidean geometry, especially for datasets with highly non-Euclidean latent anatomy. Recently, hyperbolic space has gained increasing popularity in processing graph data with tree-like structure and power-law distribution, owing to its exponential growth property. In this survey, we comprehensively revisit the technical details of the current hyperbolic graph neural networks, unifying them into a general framework and summarizing the variants of each component. More importantly, we present various HGNN-related applications. Last, we also identify several challenges, which potentially serve as guidelines for further flourishing the achievements of graph learning in hyperbolic spaces.

Recommender system is one of the most important information services on today's Internet. Recently, graph neural networks have become the new state-of-the-art approach of recommender systems. In this survey, we conduct a comprehensive review of the literature in graph neural network-based recommender systems. We first introduce the background and the history of the development of both recommender systems and graph neural networks. For recommender systems, in general, there are four aspects for categorizing existing works: stage, scenario, objective, and application. For graph neural networks, the existing methods consist of two categories, spectral models and spatial ones. We then discuss the motivation of applying graph neural networks into recommender systems, mainly consisting of the high-order connectivity, the structural property of data, and the enhanced supervision signal. We then systematically analyze the challenges in graph construction, embedding propagation/aggregation, model optimization, and computation efficiency. Afterward and primarily, we provide a comprehensive overview of a multitude of existing works of graph neural network-based recommender systems, following the taxonomy above. Finally, we raise discussions on the open problems and promising future directions of this area. We summarize the representative papers along with their codes repositories in //github.com/tsinghua-fib-lab/GNN-Recommender-Systems.

Over the past few years, we have seen fundamental breakthroughs in core problems in machine learning, largely driven by advances in deep neural networks. At the same time, the amount of data collected in a wide array of scientific domains is dramatically increasing in both size and complexity. Taken together, this suggests many exciting opportunities for deep learning applications in scientific settings. But a significant challenge to this is simply knowing where to start. The sheer breadth and diversity of different deep learning techniques makes it difficult to determine what scientific problems might be most amenable to these methods, or which specific combination of methods might offer the most promising first approach. In this survey, we focus on addressing this central issue, providing an overview of many widely used deep learning models, spanning visual, sequential and graph structured data, associated tasks and different training methods, along with techniques to use deep learning with less data and better interpret these complex models --- two central considerations for many scientific use cases. We also include overviews of the full design process, implementation tips, and links to a plethora of tutorials, research summaries and open-sourced deep learning pipelines and pretrained models, developed by the community. We hope that this survey will help accelerate the use of deep learning across different scientific domains.

Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.