查看更多>>摘要：Graph learning,when used as a semi-supervised learning(SSL)method,performs well for classification tasks with a low label rate.We provide a graph-based batch active learn-ing pipeline for pixel/patch neighborhood multi-or hyperspectral image segmentation.Our batch active learning approach selects a collection of unlabeled pixels that satisfy a graph local maximum constraint for the active learning acquisition function that determines the relative importance of each pixel to the classification.This work builds on recent advances in the design of novel active learning acquisition functions(e.g.,the Model Change approach in arXiv:2110.07739)while adding important further developments including patch-neighborhood image analysis and batch active learning methods to further increase the accuracy and greatly increase the computational efficiency of these methods.In addi-tion to improvements in the accuracy,our approach can greatly reduce the number of labeled pixels needed to achieve the same level of the accuracy based on randomly selected labeled pixels.
查看更多>>摘要：An improved algorithm for computing multiphase flows is presented in which the multi-material Moment-of-Fluid(MOF)algorithm for multiphase flows,initially described by Li et al.(2015),is enhanced addressing existing MOF difficulties in computing solutions to problems in which surface tension forces are crucial for understanding salient flow mech-anisms.The Continuous MOF(CMOF)method is motivated in this article.The CMOF reconstruction method inherently removes the"checkerboard instability"that persists when using the MOF method on surface tension driven multiphase(multimaterial)flows.The CMOF reconstruction algorithm is accelerated by coupling the CMOF method to the level set method and coupling the CMOF method to a decision tree machine learning(ML)algo-rithm.Multiphase flow examples are shown in the two-dimensional(2D),three-dimen-sional(3D)axisymmetric"RZ",and 3D coordinate systems.Examples include two mate-rial and three material multiphase flows:bubble formation,the impingement of a liquid jet on a gas bubble in a cryogenic fuel tank,freezing,and liquid lens dynamics.
查看更多>>摘要：We investigate the following inverse problem:starting from the acoustic wave equation,reconstruct a piecewise constant passive acoustic source from a single boundary temporal measurement without knowing the speed of sound.When the amplitudes of the source are known a priori,we prove a unique determination result of the shape and propose a level set algorithm to reconstruct the singularities.When the singularities of the source are known a priori,we show unique determination of the source amplitudes and propose a least-squares fitting algorithm to recover the source amplitudes.The analysis bridges the low-frequency source inversion problem and the inverse problem of gravimetry.The proposed algorithms are validated and quantitatively evaluated with numerical experiments in 2D and 3D.
查看更多>>摘要：Many important problems in science and engineering require solving the so-called para-metric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order mod-eling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the offline stage.These methods often need a pre-defined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Uti-lizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-train-ing stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster con-vergence speed without losing the accuracy than other deep learning-based methods.
查看更多>>摘要：We propose a new framework for the sampling,compression,and analysis of distributions of point sets and other geometric objects embedded in Euclidean spaces.Our approach involves constructing a tensor called the RaySense sketch,which captures nearest neigh-bors from the underlying geometry of points along a set of rays.We explore various opera-tions that can be performed on the RaySense sketch,leading to different properties and potential applications.Statistical information about the data set can be extracted from the sketch,independent of the ray set.Line integrals on point sets can be efficiently computed using the sketch.We also present several examples illustrating applications of the proposed strategy in practical scenarios.
查看更多>>摘要：We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a Riemannian version of the Adam algo-rithm.We show numerical simulations of these algorithms on various benchmarks.
查看更多>>摘要：We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embed-ding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.
查看更多>>摘要：We present the viewpoint that optimization problems encountered in machine learning can often be interpreted as minimizing a convex functional over a function space,but with a non-convex constraint set introduced by model parameterization.This observation allows us to repose such problems via a suitable relaxation as convex optimization problems in the space of distributions over the training parameters.We derive some simple relationships between the distribution-space problem and the original problem,e.g.,a distribution-space solution is at least as good as a solution in the original space.Moreover,we develop a numerical algorithm based on mixture distributions to perform approximate optimization directly in the distribution space.Consistency of this approximation is established and the numerical efficacy of the proposed algorithm is illustrated in simple examples.In both the-ory and practice,this formulation provides an alternative approach to large-scale optimiza-tion in machine learning.
查看更多>>摘要：This paper develops and analyzes a stochastic derivative-free optimization strategy.A key feature is the state-dependent adaptive variance.We prove global convergence in probability with alge-braic rate and give the quantitative results in numerical examples.A striking fact is that conver-gence is achieved without explicit information of the gradient and even without comparing differ-ent objective function values as in established methods such as the simplex method and simulated annealing.It can otherwise be compared to annealing with state-dependent temperature.
查看更多>>摘要：Active learning in semi-supervised classification involves introducing additional labels for unlabelled data to improve the accuracy of the underlying classifier.A challenge is to iden-tify which points to label to best improve performance while limiting the number of new labels."Model Change"active learning quantifies the resulting change incurred in the clas-sifier by introducing the additional label(s).We pair this idea with graph-based semi-super-vised learning(SSL)methods,that use the spectrum of the graph Laplacian matrix,which can be truncated to avoid prohibitively large computational and storage costs.We consider a family of convex loss functions for which the acquisition function can be efficiently approximated using the Laplace approximation of the posterior distribution.We show a variety of multiclass examples that illustrate improved performance over prior state-of-art.
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