Grid-free Monte Carlo methods based on the \emph{walk on spheres (WoS)} algorithm solve fundamental partial differential equations (PDEs) like the Poisson equation without discretizing the problem domain, nor approximating functions in a finite basis. Such methods hence avoid aliasing in the solution, and evade the many challenges of mesh generation. Yet for problems with complex geometry, practical grid-free methods have been largely limited to basic Dirichlet boundary conditions. This paper introduces the \emph{walk on stars (WoSt)} method, which solves linear elliptic PDEs with arbitrary mixed Neumann and Dirichlet boundary conditions. The key insight is that one can efficiently simulate reflecting Brownian motion (which models Neumann conditions) by replacing the balls used by WoS with \emph{star-shaped} domains; we identify such domains by locating the closest visible point on the geometric silhouette. Overall, WoSt retains many attractive features of other grid-free Monte Carlo methods, such as progressive evaluation, trivial parallel implementation, and logarithmic scaling relative to geometric complexity.
相關內容
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to the second-order elliptic PDE in non-divergence form. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the 2D and 3D settings. In addition, we present a comparison with the existing multivariate spline methods in \cite{ALW06} and \cite{LW17} to show that the new method produces a similar and sometimes more accurate approximation in a more efficient fashion.
We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of polynomials of degree N in space. We prove that we can choose a sequence of discrete controls depending on the parameter N associated with the approximate control problem in such a way that they converge, as N goes to infinity, to a control of the continuous wave equation. Unlike other numerical approximations tried in the literature, this one does not require regularization techniques and can be easily adapted to other equations and systems where the controllability of the continuous model is known. The method is illustrated with several examples in 1-d and 2-d in a square domain. We also give numerical evidence of the highly accurate approximation inherent to spectral methods.
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.
In the last decade or so, we have witnessed deep learning reinvigorating the machine learning field. It has solved many problems in the domains of computer vision, speech recognition, natural language processing, and various other tasks with state-of-the-art performance. The data is generally represented in the Euclidean space in these domains. Various other domains conform to non-Euclidean space, for which graph is an ideal representation. Graphs are suitable for representing the dependencies and interrelationships between various entities. Traditionally, handcrafted features for graphs are incapable of providing the necessary inference for various tasks from this complex data representation. Recently, there is an emergence of employing various advances in deep learning to graph data-based tasks. This article provides a comprehensive survey of graph neural networks (GNNs) in each learning setting: supervised, unsupervised, semi-supervised, and self-supervised learning. Taxonomy of each graph based learning setting is provided with logical divisions of methods falling in the given learning setting. The approaches for each learning task are analyzed from both theoretical as well as empirical standpoints. Further, we provide general architecture guidelines for building GNNs. Various applications and benchmark datasets are also provided, along with open challenges still plaguing the general applicability of GNNs.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.
A key requirement for the success of supervised deep learning is a large labeled dataset - a condition that is difficult to meet in medical image analysis. Self-supervised learning (SSL) can help in this regard by providing a strategy to pre-train a neural network with unlabeled data, followed by fine-tuning for a downstream task with limited annotations. Contrastive learning, a particular variant of SSL, is a powerful technique for learning image-level representations. In this work, we propose strategies for extending the contrastive learning framework for segmentation of volumetric medical images in the semi-supervised setting with limited annotations, by leveraging domain-specific and problem-specific cues. Specifically, we propose (1) novel contrasting strategies that leverage structural similarity across volumetric medical images (domain-specific cue) and (2) a local version of the contrastive loss to learn distinctive representations of local regions that are useful for per-pixel segmentation (problem-specific cue). We carry out an extensive evaluation on three Magnetic Resonance Imaging (MRI) datasets. In the limited annotation setting, the proposed method yields substantial improvements compared to other self-supervision and semi-supervised learning techniques. When combined with a simple data augmentation technique, the proposed method reaches within 8% of benchmark performance using only two labeled MRI volumes for training, corresponding to only 4% (for ACDC) of the training data used to train the benchmark.
Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics, for decades. Nowadays, estimating causal effect from observational data has become an appealing research direction owing to the large amount of available data and low budget requirement, compared with randomized controlled trials. Embraced with the rapidly developed machine learning area, various causal effect estimation methods for observational data have sprung up. In this survey, we provide a comprehensive review of causal inference methods under the potential outcome framework, one of the well known causal inference framework. The methods are divided into two categories depending on whether they require all three assumptions of the potential outcome framework or not. For each category, both the traditional statistical methods and the recent machine learning enhanced methods are discussed and compared. The plausible applications of these methods are also presented, including the applications in advertising, recommendation, medicine and so on. Moreover, the commonly used benchmark datasets as well as the open-source codes are also summarized, which facilitate researchers and practitioners to explore, evaluate and apply the causal inference methods.
Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.
Lots of learning tasks require dealing with graph data which contains rich relation information among elements. Modeling physics system, learning molecular fingerprints, predicting protein interface, and classifying diseases require that a model to learn from graph inputs. In other domains such as learning from non-structural data like texts and images, reasoning on extracted structures, like the dependency tree of sentences and the scene graph of images, is an important research topic which also needs graph reasoning models. Graph neural networks (GNNs) are connectionist models that capture the dependence of graphs via message passing between the nodes of graphs. Unlike standard neural networks, graph neural networks retain a state that can represent information from its neighborhood with an arbitrary depth. Although the primitive graph neural networks have been found difficult to train for a fixed point, recent advances in network architectures, optimization techniques, and parallel computation have enabled successful learning with them. In recent years, systems based on graph convolutional network (GCN) and gated graph neural network (GGNN) have demonstrated ground-breaking performance on many tasks mentioned above. In this survey, we provide a detailed review over existing graph neural network models, systematically categorize the applications, and propose four open problems for future research.
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