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The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and, at the same time, decouples space and velocity discretizations. In this study, we investigate the numerical convergence of MLBM in two benchmark tests: the Taylor-Green vortex and annular (bent) channel flow. We compare our MLBM results to LBM and to the analytical solution of the Navier-Stokes equation. We investigate the method's convergence in terms of the discretization parameter, the interpolation order, and the LBM streaming distance refinement. We observe that MLBM outperforms LBM in terms of the error value for the same number of nodes discretizing the domain. We find that LBM errors at a given streaming distance $\delta x$ and timestep length $\delta t$ are the asymptotic lower bounds of MLBM errors with the same streaming distance and timestep length. Finally, we suggest an expression for the MLBM error that consists of the LBM error and other terms related to the semi-Lagrangian nature of the discussed method itself.

In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PDE systems can be reformulated as WHFs. We use parameterized function as push-forward map to characterize the solution of WHF, and convert the PDE to a finite-dimensional ODE system, which is a Hamiltonian system in the phase space of the parameter manifold. We establish error analysis results for the continuous time approximation scheme in Wasserstein metric. For the numerical implementation, we use neural networks as push-forward maps. We apply an effective symplectic scheme to solve the derived Hamiltonian ODE system so that the method preserves some important quantities such as total energy. The computation is done by fully deterministic symplectic integrator without any neural network training. Thus, our method does not involve direct optimization over network parameters and hence can avoid the error introduced by stochastic gradient descent (SGD) methods, which is usually hard to quantify and measure. The proposed algorithm is a sampling-based approach that scales well to higher dimensional problems. In addition, the method also provides an alternative connection between the Lagrangian and Eulerian perspectives of the original WHF through the parameterized ODE dynamics.

In this work, we develop an efficient high order discontinuous Galerkin (DG) method for solving the Electrical Impedance Tomography (EIT). EIT is a highly nonlinear ill-posed inverse problem where the interior conductivity of an object is recovered from the surface measurements of voltage and current flux. We first propose a new optimization problem based on the recovery of the conductivity from the Dirichlet-to-Neumann map to minimize the mismatch between the predicted current and the measured current on the boundary. And we further prove the existence of the minimizer. Numerically the optimization problem is solved by a third order DG method with quadratic polynomials. Numerical results for several two-dimensional problems with both single and multiple inclusions are demonstrated to show the high {accuracy and efficiency} of the proposed high order DG method. Analysis and computation for discontinuous conductivities are also studied in this work.

Large discrete action spaces remain a central challenge for reinforcement learning methods. Such spaces are encountered in many real-world applications, e.g., recommender systems, multi-step planning, and inventory replenishment. The mapping of continuous proxies to discrete actions is a promising paradigm for handling large discrete action spaces. Existing continuous-to-discrete mapping approaches involve searching for discrete neighboring actions in a static pre-defined neighborhood, which requires discrete neighbor lookups across the entire action space. Hence, scalability issues persist. To mitigate this drawback, we propose a novel Dynamic Neighborhood Construction (DNC) method, which dynamically constructs a discrete neighborhood to map the continuous proxy, thus efficiently exploiting the underlying action space. We demonstrate the robustness of our method by benchmarking it against three state-of-the-art approaches designed for large discrete action spaces across three different environments. Our results show that DNC matches or outperforms state-of-the-art approaches while being more computationally efficient. Furthermore, our method scales to action spaces that so far remained computationally intractable for existing methodologies.

In this paper we discuss potentially practical ways to produce expander graphs with good spectral properties and a compact description. We focus on several classes of uniform and bipartite expander graphs defined as random Schreier graphs of the general linear group over the finite field of size two. We perform numerical experiments and show that such constructions produce spectral expanders that can be useful for practical applications. To find a theoretical explanation of the observed experimental results, we used the method of moments to prove upper bounds for the expected second largest eigenvalue of the random Schreier graphs used in our constructions. We focus on bounds for which it is difficult to study the asymptotic behaviour but it is possible to compute non-trivial conclusions for relatively small graphs with parameters from our numerical experiments (e.g., with less than 2^200 vertices and degree at least logarithmic in the number of vertices).

Accurate segmentation of large areas from very high spatial-resolution (VHR) remote sensing imagery remains a challenging issue in image analysis. Existing supervised and unsupervised methods both suffer from the large variance of object sizes and the difficulty in scale selection, which often result in poor segmentation accuracies. To address the above challenges, we propose a deep learning-based region-merging method (DeepMerge) to handle the segmentation in large VHR images by integrating a Transformer with a multi-level embedding module, a segment-based feature embedding module and a region-adjacency graph model. In addition, we propose a modified binary tree sampling method to generate multi-level inputs from initial segmentation results, serving as inputs for the DeepMerge model. To our best knowledge, the proposed method is the first to use deep learning to learn the similarity between adjacent segments for region-merging. The proposed DeepMerge method is validated using a remote sensing image of 0.55m resolution covering an area of 5,660 km^2 acquired from Google Earth. The experimental results show that the proposed DeepMerge with the highest F value (0.9446) and the lowest TE (0.0962) and ED2 (0.8989) is able to correctly segment objects of different sizes and outperforms all selected competing segmentation methods from both quantitative and qualitative assessments.

The energy dissipation law and maximum bound principle are significant characteristics of the Allen-Chan equation. To preserve discrete counterpart of these properties, the linear part of the target system is usually discretized implicitly, resulting in a large linear or nonlinear system of equations. The Fast Fourier Transform (FFT) algorithm is commonly used to solve the resulting linear or nonlinear systems with computational costs of $\mathcal{O}(M^d log M)$ at each time step, where $M$ is the number of spatial grid points in each direction, and $d$ is the dimension of the problem. Combining the Saul'yev methods and the stabilized technique, we propose and analyze novel first- and second-order numerical schemes for the Allen-Cahn equation in this paper. In contrast to the traditional methods, the proposed methods can be solved by components, requiring only $\mathcal{O}(M^d)$ computational costs per time step. Additionally, they preserve the maximum bound principle and original energy dissipation law at the discrete level. We also propose rigorous analysis of their consistency and convergence. Numerical experiments are conducted to confirm the theoretical analysis and demonstrate the efficiency of the proposed methods.

The important phenomenon of "stickiness" of chaotic orbits in low dimensional dynamical systems has been investigated for several decades, in view of its applications to various areas of physics, such as classical and statistical mechanics, celestial mechanics and accelerator dynamics. Most of the work to date has focused on two-degree of freedom Hamiltonian models often represented by two-dimensional (2D) area preserving maps. In this paper, we extend earlier results using a 4-dimensional extension of the 2D MacMillan map, and show that a symplectic model of two coupled MacMillan maps also exhibits stickiness phenomena in limited regions of phase space. To this end, we employ probability distributions in the sense of the Central Limit Theorem to demonstrate that, as in the 2D case, sticky regions near the origin are also characterized by "weak" chaos and Tsallis entropy, in sharp contrast to the "strong" chaos that extends over much wider domains and is described by Boltzmann Gibbs statistics. Remarkably, similar stickiness phenomena have been observed in higher dimensional Hamiltonian systems around unstable simple periodic orbits at various values of the total energy of the system.

Graph convolutional neural networks have recently shown great potential for the task of zero-shot learning. 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, multi-layer architectures, which are required to propagate knowledge to distant nodes in the graph, dilute the knowledge by performing extensive Laplacian smoothing at each layer and thereby consequently decrease performance. In order to still enjoy the benefit brought by the graph structure while preventing dilution of knowledge from distant nodes, we propose a Dense Graph Propagation (DGP) module with carefully designed direct links among distant nodes. DGP 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. A weighting scheme is further used to weigh their contribution depending on the distance to the node to improve information propagation in the graph. Combined with finetuning of the representations in a two-stage training approach our method outperforms state-of-the-art zero-shot learning approaches.

Multi-view networks are ubiquitous in real-world applications. In order to extract knowledge or business value, it is of interest to transform such networks into representations that are easily machine-actionable. Meanwhile, network embedding has emerged as an effective approach to generate distributed network representations. Therefore, we are motivated to study the problem of multi-view network embedding, with a focus on the characteristics that are specific and important in embedding this type of networks. In our practice of embedding real-world multi-view networks, we identify two such characteristics, which we refer to as preservation and collaboration. We then explore the feasibility of achieving better embedding quality by simultaneously modeling preservation and collaboration, and propose the mvn2vec algorithms. With experiments on a series of synthetic datasets, an internal Snapchat dataset, and two public datasets, we further confirm the presence and importance of preservation and collaboration. These experiments also demonstrate that better embedding can be obtained by simultaneously modeling the two characteristics, while not over-complicating the model or requiring additional supervision.