In this paper, we present a hybrid neural-network and MAC (Marker-And-Cell) scheme for solving Stokes equations with singular forces on an embedded interface in regular domains. As known, the solution variables (the pressure and velocity) exhibit non-smooth behaviors across the interface so extra discretization efforts must be paid near the interface in order to have small order of local truncation errors in finite difference schemes. The present hybrid approach avoids such additional difficulty. It combines the expressive power of neural networks with the convergence of finite difference schemes to ease the code implementation and to achieve good accuracy at the same time. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular part solution, while the standard MAC scheme is used to obtain the regular part solution with associated boundary conditions. The two- and three-dimensional numerical results show that the present hybrid method converges with second-order accuracy for the velocity and first-order accuracy for the pressure, and it is comparable with the traditional immersed interface method in literature.
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We present a new technique for visualizing high-dimensional data called cluster MDS (cl-MDS), which addresses a common difficulty of dimensionality reduction methods: preserving both local and global structures of the original sample in a single 2-dimensional visualization. Its algorithm combines the well-known multidimensional scaling (MDS) tool with the $k$-medoids data clustering technique, and enables hierarchical embedding, sparsification and estimation of 2-dimensional coordinates for additional points. While cl-MDS is a generally applicable tool, we also include specific recipes for atomic structure applications. We apply this method to non-linear data of increasing complexity where different layers of locality are relevant, showing a clear improvement in their retrieval and visualization quality.
In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in subdomains bounded by discontinuity lines (DLs), the latter defined from the Rankine-Hugoniot jump conditions. The proposed approach allows to efficiently consider an arbitrary number of entropic shock waves, shock wave generation, as well as wave interactions. Some numerical experiments are presented to illustrate the strengths and properties of the algorithms.
In this work we present two new numerical schemes to approximate the Navier-Stokes-Cahn-Hilliard system with degenerate mobility using finite differences in time and finite elements in space. The proposed schemes are conservative, energy-stable and preserve the maximum principle approximately (the amount of the phase variable being outside of the interval [0,1] goes to zero in terms of a truncation parameter). Additionally, we present several numerical results to illustrate the accuracy and the well behavior of the proposed schemes, as well as a comparison with the behavior of the Navier-Stokes-Cahn-Hilliard model with constant mobility.
In this work, we consider the numerical computation of ground states and dynamics of single-component Bose-Einstein condensates (BECs). The corresponding models are spatially discretized with a multiscale finite element approach known as Localized Orthogonal Decomposition (LOD). Despite the outstanding approximation properties of such a discretization in the context of BECs, taking full advantage of it without creating severe computational bottlenecks can be tricky. In this paper, we therefore present two fully-discrete numerical approaches that are formulated in such a way that they take special account of the structure of the LOD spaces. One approach is devoted to the computation of ground states and another one for the computation of dynamics. A central focus of this paper is also the discussion of implementation aspects that are very important for the practical realization of the methods. In particular, we discuss the use of suitable data structures that keep the memory costs economical. The paper concludes with various numerical experiments in 1d, 2d and 3d that investigate convergence rates and approximation properties of the methods and which demonstrate their performance and computational efficiency, also in comparison to spectral and standard finite element approaches.
Existing approaches for device placement ignore the topological features of computation graphs and rely mostly on heuristic methods for graph partitioning. At the same time, they either follow a grouper-placer or an encoder-placer architecture, which requires understanding the interaction structure between code operations. To bridge the gap between encoder-placer and grouper-placer techniques, we propose a novel framework for the task of device placement, relying on smaller computation graphs extracted from the OpenVINO toolkit using reinforcement learning. The framework consists of five steps, including graph coarsening, node representation learning and policy optimization. It facilitates end-to-end training and takes into consideration the directed and acyclic nature of the computation graphs. We also propose a model variant, inspired by graph parsing networks and complex network analysis, enabling graph representation learning and personalized graph partitioning jointly, using an unspecified number of groups. To train the entire framework, we utilize reinforcement learning techniques by employing the execution time of the suggested device placements to formulate the reward. We demonstrate the flexibility and effectiveness of our approach through multiple experiments with three benchmark models, namely Inception-V3, ResNet, and BERT. The robustness of the proposed framework is also highlighted through an ablation study. The suggested placements improve the inference speed for the benchmark models by up to $58.2\%$ over CPU execution and by up to $60.24\%$ compared to other commonly used baselines.
In this paper, we propose R\'enyi information generating function (RIGF) and discuss its various properties. The relation between the RIGF and Shannon entropy of order $q>0$ is established. Several bounds are obtained. The RIGF of escort distribution is also derived. Furthermore, we introduce R\'enyi divergence information generating function (RDIGF) and discuss its effect under monotone transformations. Next, we propose Jensen-R\'enyi information generating function (JRIGF) and establish its properties. In addition, we present non-parametric and parametric estimators of the RIGF. For illustrative purpose, a simulation study is carried out and a real data relating to the failure times of electronic components is analyzed. Finally, a comparison study between the non-parametric and parametric estimators is made in terms of absolute bias and mean square error (MSE).
In this paper we analyze a conforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. We consider the classic velocity-pressure formulation which allows us to consider the divergence-conforming virtual element spaces employed for the Stokes equations. Under standard assumptions on the meshes we derive a priori error estimates for the proposed method with the aid of the compact operators theory. We report some numerical tests to confirm the theoretical results.
The largest eigenvalue of the Hessian, or sharpness, of neural networks is a key quantity to understand their optimization dynamics. In this paper, we study the sharpness of deep linear networks for overdetermined univariate regression. Minimizers can have arbitrarily large sharpness, but not an arbitrarily small one. Indeed, we show a lower bound on the sharpness of minimizers, which grows linearly with depth. We then study the properties of the minimizer found by gradient flow, which is the limit of gradient descent with vanishing learning rate. We show an implicit regularization towards flat minima: the sharpness of the minimizer is no more than a constant times the lower bound. The constant depends on the condition number of the data covariance matrix, but not on width or depth. This result is proven both for a small-scale initialization and a residual initialization. Results of independent interest are shown in both cases. For small-scale initialization, we show that the learned weight matrices are approximately rank-one and that their singular vectors align. For residual initialization, convergence of the gradient flow for a Gaussian initialization of the residual network is proven. Numerical experiments illustrate our results and connect them to gradient descent with non-vanishing learning rate.
In this paper, we propose a weak Galerkin (WG) finite element method for the Maxwell eigenvalue problem. By restricting subspaces, we transform the mixed form of Maxwell eigenvalue problem into simple elliptic equation. Then we give the WG numerical scheme for the Maxwell eigenvalue problem. Furthermore, we obtain the optimal error estimates of arbitrarily high convergence order and prove the lower bound property of numerical solutions for eigenvalues. Numerical experiments show the accuracy of theoretical analysis and the property of lower bound.
Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.
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