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With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(\Omega)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method., Comment: 32 pages, 10 figures, to appear at SIAM Journal on Imaging Sciences

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with $\mathrm{ReLU}^{3}$ networks (deep network with activations function $\max\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with $\mathrm{ReLU}^{3}$ network, which is of immense independent interest., arXiv admin note: text overlap with arXiv:2103.13330

In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $\omega_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(\omega_f(\sqrt{d})\cdot2^{-M}+\omega_{f}\left(\frac{\sqrt{d}}{N}\right))$, where $M,N\in \mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $\max\left\{\left\lceil2d^{3/2}\left(\frac{3\mu}{\epsilon}\right)^{1/{\alpha}}\right\rceil,2\left\lceil\log_2\frac{3\mu d^{\alpha/2}}{2\epsilon}\right\rceil+2\right\}$ that approximates $f\in \mathcal{H}_{\mu}^{\alpha}([0,1]^d)$ within a given tolerance $\epsilon >0$ measured in $L^p$ norm $p\in[1,\infty)$, where $\mathcal{H}_{\mu}^{\alpha}([0,1]^d)$ denotes the H\"older continuous function class defined on $[0,1]^d$ with order $\alpha \in (0,1]$ and constant $\mu > 0$. Therefore, the ReLU-sine-$2^x$ networks overcome the curse of dimensionality on $\mathcal{H}_{\mu}^{\alpha}([0,1]^d)$. In addition to its supper expressive power, functions implemented by ReLU-sine-$2^x$ networks are (generalized) differentiable, enabling us to apply SGD to train.

We propose an Euler particle transport (EPT) approach for generative learning. The proposed approach is motivated by the problem of finding an optimal transport map from a reference distribution to a target distribution characterized by the Monge-Ampere equation. Interpreting the infinitesimal linearization of the Monge-Ampere equation from the perspective of gradient flows in measure spaces leads to a stochastic McKean-Vlasov equation. We use the forward Euler method to solve this equation. The resulting forward Euler map pushes forward a reference distribution to the target. This map is the composition of a sequence of simple residual maps, which are computationally stable and easy to train. The key task in training is the estimation of the density ratios or differences that determine the residual maps. We estimate the density ratios (differences) based on the Bregman divergence with a gradient penalty using deep density-ratio (difference) fitting. We show that the proposed density-ratio (difference) estimators do not suffer from the "curse of dimensionality" if data is supported on a lower-dimensional manifold. Numerical experiments with multi-mode synthetic datasets and comparisons with the existing methods on real benchmark datasets support our theoretical results and demonstrate the effectiveness of the proposed method., arXiv admin note: substantial text overlap with arXiv:2002.02862

The main goal of 1-bit compressive sampling is to decode $n$ dimensional signals with sparsity level $s$ from $m$ binary measurements. This is a challenging task due to the presence of nonlinearity, noises and sign flips. In this paper, the cardinality constraint least square is proposed as a desired decoder. We prove that, up to a constant $c$, with high probability, the proposed decoder achieves a minimax estimation error as long as $m \geq \mathcal{O}( s\log n)$. Computationally, we utilize a generalized Newton algorithm (GNA) to solve the cardinality constraint minimization problem with the cost of solving a least squares problem with small size at each iteration. We prove that, with high probability, the $\ell_{\infty}$ norm of the estimation error between the output of GNA and the underlying target decays to $\mathcal{O}(\sqrt{\frac{\log n }{m}}) $ after at most $\mathcal{O}(\log s)$ iterations. Moreover, the underlying support can be recovered with high probability in $\mathcal{O}(\log s)$ steps provided that the target signal is detectable. Extensive numerical simulations and comparisons with state-of-the-art methods are presented to illustrate the robustness of our proposed decoder and the efficiency of the GNA algorithm., arXiv admin note: text overlap with arXiv:1711.01206

Screening and working set techniques are important approaches to reducing the size of an optimization problem. They have been widely used in accelerating first-order methods for solving large-scale sparse learning problems. In this paper, we develop a new screening method called Newton screening (NS) which is a generalized Newton method with a built-in screening mechanism. We derive an equivalent KKT system for the Lasso and utilize a generalized Newton method to solve the KKT equations. Based on this KKT system, a built-in working set with a relatively small size is first determined using the sum of primal and dual variables generated from the previous iteration, then the primal variable is updated by solving a least-squares problem on the working set and the dual variable updated based on a closed-form expression. Moreover, we consider a sequential version of Newton screening (SNS) with a warm-start strategy. We show that NS possesses an optimal convergence property in the sense that it achieves one-step local convergence. Under certain regularity conditions on the feature matrix, we show that SNS hits a solution with the same signs as the underlying true target and achieves a sharp estimation error bound with high probability. Simulation studies and real data analysis support our theoretical results and demonstrate that SNS is faster and more accurate than several state-of-the-art methods in our comparative studies., This paper has fatal mistakes

We propose a semismooth Newton algorithm for pathwise optimization (SNAP) for the LASSO and Enet in sparse, high-dimensional linear regression. SNAP is derived from a suitable formulation of the KKT conditions based on Newton derivatives. It solves the semismooth KKT equations efficiently by actively and continuously seeking the support of the regression coefficients along the solution path with warm start. At each knot in the path, SNAP converges locally superlinearly for the Enet criterion and achieves an optimal local convergence rate for the LASSO criterion, i.e., SNAP converges in one step at the cost of two matrix-vector multiplication per iteration. Under certain regularity conditions on the design matrix and the minimum magnitude of the nonzero elements of the target regression coefficients, we show that SNAP hits a solution with the same signs as the regression coefficients and achieves a sharp estimation error bound in finite steps with high probability. The computational complexity of SNAP is shown to be the same as that of LARS and coordinate descent algorithms per iteration. Simulation studies and real data analysis support our theoretical results and demonstrate that SNAP is faster and accurate than LARS and coordinate descent algorithms.

Destriping is a classical problem in remote sensing image processing. Although considerable effort has been made to remove stripes, few of the existing methods can eliminate stripe noise with arbitrary orientations. This situation makes the removal of oblique stripes in the higher-level remote sensing products become an unfinished and urgent issue. To overcome the challenging problem, we propose a novel destriping model which is self-adjusted to different orientations of stripe noise. First of all, the oriented variation model is designed to accomplish the stripe orientation approximation. In this model, the stripe direction is automatically estimated and then imbedded into the constraint term to depict the along-stripe smoothness of the stripe component. Mainly based on the oriented variation model, a whole destriping framework is proposed by jointly employing an L1-norm constraint and a TV regularization to separately capture the global distribution property of stripe component and the piecewise smoothness of the clean image. The qualitative and quantitative experimental results of both orientation and destriping aspects confirm the effectiveness and stability of the proposed method., 14 pages, 14 figures

Stochastic gradient descent is one of the most successful approaches for solving large-scale problems, especially in machine learning and statistics. At each iteration, it employs an unbiased estimator of the full gradient computed from one single randomly selected data point. Hence, it scales well with problem size and is very attractive for truly massive dataset, and holds significant potentials for solving large-scale inverse problems. In the recent literature of machine learning, it was empirically observed that when equipped with early stopping, it has regularizing property. In this work, we rigorously establish its regularizing property (under \textit{a priori} early stopping rule), and also prove convergence rates under the canonical sourcewise condition, for minimizing the quadratic functional for linear inverse problems. This is achieved by combining tools from classical regularization theory and stochastic analysis. Further, we analyze the preasymptotic weak and strong convergence behavior of the algorithm. The theoretical findings shed insights into the performance of the algorithm, and are complemented with illustrative numerical experiments., 22 pages, better presentation

We develop a primal dual active set with continuation algorithm for solving the \ell^0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly. Under certain conditions on the sensing matrix, i.e., mutual incoherence property or restricted isometry property, and the noise level, the finite step global convergence of the algorithm is established. Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and the convergence analysis.