arXiv subjects

Zhu Wang

Publications and source records attributed to Zhu Wang.

At least 55 records · Page 3Linked to original sources

arXiv2022

Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver

Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional numerical methods have been developed and widely used for their solutions. Inspired by rapidly growing impact of deep learning on scientific and engineering research, in this paper we propose a novel neural network, GF-Net, for learning the Green's functions of linear reaction-diffusion equations in an unsupervised fashion. The proposed method overcomes the challenges for finding the Green's functions of the equations on arbitrary domains by utilizing physics-informed approach and the symmetry of the Green's function. As a consequence, it particularly leads to an efficient way for solving the target equations under different boundary conditions and sources. We also demonstrate the effectiveness of the proposed approach by experiments in square, annular and L-shape domains.

cs.LG↗

arXiv2022

Zero-Shot Multi-Modal Artist-Controlled Retrieval and Exploration of 3D Object Sets

When creating 3D content, highly specialized skills are generally needed to design and generate models of objects and other assets by hand. We address this problem through high-quality 3D asset retrieval from multi-modal inputs, including 2D sketches, images and text. We use CLIP as it provides a bridge to higher-level latent features. We use these features to perform a multi-modality fusion to address the lack of artistic control that affects common data-driven approaches. Our approach allows for multi-modal conditional feature-driven retrieval through a 3D asset database, by utilizing a combination of input latent embeddings. We explore the effects of different combinations of feature embeddings across different input types and weighting methods.

cs.CV↗

arXiv2022

Estimating Atmospheric Parameters from LAMOST Low-Resolution Spectra with Low SNR

Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) acquired tens of millions of low-resolution stellar spectra. The large amount of the spectra result in the urgency to explore automatic atmospheric parameter estimation methods. There are lots of LAMOST spectra with low signal-to-noise ratios (SNR), which result in a sharp degradation on the accuracy of their estimations. Therefore, it is necessary to explore better estimation methods for low-SNR spectra. This paper proposed a neural network-based scheme to deliver atmospheric parameters, LASSO-MLPNet. Firstly, we adopt a polynomial fitting method to obtain pseudo-continuum and remove it. Then, some parameter-sensitive features in the existence of high noises were detected using Least Absolute Shrinkage and Selection Operator (LASSO). Finally, LASSO-MLPNet used a Multilayer Perceptron network (MLPNet) to estimate atmospheric parameters $T_{\mathrm{eff}}$, log $g$ and [Fe/H]. The effectiveness of the LASSO-MLPNet was evaluated on some LAMOST stellar spectra of the common star between APOGEE (The Apache Point Observatory Galactic Evolution Experiment) and LAMOST. it is shown that the estimation accuracy is significantly improved on the stellar spectra with $10<\mathrm{SNR}\leq80$. Especially, LASSO-MLPNet reduces the mean absolute error (MAE) of the estimation of $T_{\mathrm{eff}}$, log $g$ and [Fe/H] from (144.59 K, 0.236 dex, 0.108 dex) (LASP) to (90.29 K, 0.152 dex, 0.064 dex) (LASSO-MLPNet) on the stellar spectra with $10<\mathrm{SNR}\leq20$. To facilitate reference, we release the estimates of the LASSO-MLPNet from more than 4.82 million stellar spectra with $10<\mathrm{SNR}\leq80$ and 3500 < SNR$g$ $\leq$ 6500 as a value-added output.

astro-ph.IM↗

arXiv2022

Nonlinear Reduced DNN Models for State Estimation

We propose in this paper a data driven state estimation scheme for generating nonlinear reduced models for parametric families of PDEs, directly providing data-to-state maps, represented in terms of Deep Neural Networks. A major constituent is a sensor-induced decomposition of a model-compliant Hilbert space warranting approximation in problem relevant metrics. It plays a similar role as in a Parametric Background Data Weak framework for state estimators based on Reduced Basis concepts. Extensive numerical tests shed light on several optimization strategies that are to improve robustness and performance of such estimators.

math.NA↗

arXiv2022

Estimation of stellar atmospheric parameters from LAMOST DR8 low-resolution spectra with 20$\leq$SNR$<$30

The accuracy of the estimated stellar atmospheric parameter decreases evidently with the decreasing of spectral signal-to-noise ratio (SNR) and there are a huge amount of this kind observations, especially in case of SNR$<$30. Therefore, it is helpful to improve the parameter estimation performance for these spectra and this work studied the ($T_\texttt{eff}, \log~g$, [Fe/H]) estimation problem for LAMOST DR8 low-resolution spectra with 20$\leq$SNR$<$30. We proposed a data-driven method based on machine learning techniques. Firstly, this scheme detected stellar atmospheric parameter-sensitive features from spectra by the Least Absolute Shrinkage and Selection Operator (LASSO), rejected ineffective data components and irrelevant data. Secondly, a Multi-layer Perceptron (MLP) method was used to estimate stellar atmospheric parameters from the LASSO features. Finally, the performance of the LASSO-MLP was evaluated by computing and analyzing the consistency between its estimation and the reference from the APOGEE (Apache Point Observatory Galactic Evolution Experiment) high-resolution spectra. Experiments show that the Mean Absolute Errors (MAE) of $T_\texttt{eff}, \log~g$, [Fe/H] are reduced from the LASP (137.6 K, 0.195 dex, 0.091 dex) to LASSO-MLP (84.32 K, 0.137 dex, 0.063 dex), which indicate evident improvements on stellar atmospheric parameter estimation. In addition, this work estimated the stellar atmospheric parameters for 1,162,760 low-resolution spectra with 20$\leq$SNR$<$30 from LAMOST DR8 using LASSO-MLP, and released the estimation catalog, learned model, experimental code, trained model, training data and test data for scientific exploration and algorithm study.

astro-ph.GA↗

arXiv2022

MM for Penalized Estimation

Penalized estimation can conduct variable selection and parameter estimation simultaneously. The general framework is to minimize a loss function subject to a penalty designed to generate sparse variable selection. The majorization-minimization (MM) algorithm is a computational scheme for stability and simplicity, and the MM algorithm has been widely applied in penalized estimation. Much of the previous work have focused on convex loss functions such as generalized linear models. When data are contaminated with outliers, robust loss functions can generate more reliable estimates. Recent literature has witnessed a growing impact of nonconvex loss-based methods, which can generate robust estimation for data contaminated with outliers. This article investigates MM algorithm for penalized estimation, provide innovative optimality conditions and establish convergence theory with both convex and nonconvex loss functions. With respect to applications, we focus on several nonconvex loss functions, which were formerly studied in machine learning for regression and classification problems. Performance of the proposed algorithms are evaluated on simulated and real data including healthcare costs and cancer clinical status. Efficient implementations of the algorithms are available in the R package mpath in CRAN.

stat.CO↗

arXiv2022

Edge Data Based Trailer Inception Probabilistic Matrix Factorization for Context-Aware Movie Recommendation

The rapid growth of edge data generated by mobile devices and applications deployed at the edge of the network has exacerbated the problem of information overload. As an effective way to alleviate information overload, recommender system can improve the quality of various services by adding application data generated by users on edge devices, such as visual and textual information, on the basis of sparse rating data. The visual information in the movie trailer is a significant part of the movie recommender system. However, due to the complexity of visual information extraction, data sparsity cannot be remarkably alleviated by merely using the rough visual features to improve the rating prediction accuracy. Fortunately, the convolutional neural network can be used to extract the visual features precisely. Therefore, the end-to-end neural image caption (NIC) model can be utilized to obtain the textual information describing the visual features of movie trailers. This paper proposes a trailer inception probabilistic matrix factorization model called Ti-PMF, which combines NIC, recurrent convolutional neural network, and probabilistic matrix factorization models as the rating prediction model. We implement the proposed Ti-PMF model with extensive experiments on three real-world datasets to validate its effectiveness. The experimental results illustrate that the proposed Ti-PMF outperforms the existing ones.

cs.CV↗

arXiv2022

VRConvMF: Visual Recurrent Convolutional Matrix Factorization for Movie Recommendation

Sparsity of user-to-item rating data becomes one of challenging issues in the recommender systems, which severely deteriorates the recommendation performance. Fortunately, context-aware recommender systems can alleviate the sparsity problem by making use of some auxiliary information, such as the information of both the users and items. In particular, the visual information of items, such as the movie poster, can be considered as the supplement for item description documents, which helps to obtain more item features. In this paper, we focus on movie recommender system and propose a probabilistic matrix factorization based recommendation scheme called visual recurrent convolutional matrix factorization (VRConvMF), which utilizes the textual and multi-level visual features extracted from the descriptive texts and posters respectively. We implement the proposed VRConvMF and conduct extensive experiments on three commonly used real world datasets to validate its effectiveness. The experimental results illustrate that the proposed VRConvMF outperforms the existing schemes.

cs.IR↗

arXiv2022

A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modeling

The popularity of deep convolutional autoencoders (CAEs) has engendered new and effective reduced-order models (ROMs) for the simulation of large-scale dynamical systems. Despite this, it is still unknown whether deep CAEs provide superior performance over established linear techniques or other network-based methods in all modeling scenarios. To elucidate this, the effect of autoencoder architecture on its associated ROM is studied through the comparison of deep CAEs against two alternatives: a simple fully connected autoencoder, and a novel graph convolutional autoencoder. Through benchmark experiments, it is shown that the superior autoencoder architecture for a given ROM application is highly dependent on the size of the latent space and the structure of the snapshot data, with the proposed architecture demonstrating benefits on data with irregular connectivity when the latent space is sufficiently large.

cs.LG↗

arXiv2021

High-Order Multirate Explicit Time-Stepping Schemes for the Baroclinic-Barotropic Split Dynamics in Primitive Equations

In order to treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. Based on the framework of strong stability-preserving Runge-Kutta approach, we propose two high-order multirate explicit time-stepping schemes (SSPRK2-SE and SSPRK3-SE) for the resulting split system in this paper. The proposed schemes allow for a large time step to be used for the three-dimensional baroclinic (slow) mode and a small time step for the two-dimensional barotropic (fast) mode, in which each of the two mode solves just need to satisfy their respective CFL conditions for numerical stability. Specifically, at each time step, the baroclinic velocity is first computed by advancing the baroclinic mode and fluid thickness of the system with the large time-step \textcolor{black}{and the assistance of some intermediate approximations of the baroctropic mode obtained by substepping with the small-time step}; then the barotropic velocity is corrected by using the small time step to re-advance the barotropic mode under an improved barotropic forcing produced by interpolation of the forcing terms from the preceding baroclinic mode solves; lastly, the fluid thickness is updated by coupling the baroclinic and barotropic velocities. Additionally, numerical inconsistencies on the discretized sea surface height caused by the mode splitting are relieved via a reconciliation process with carefully calculated flux deficits. Two benchmark tests from the "MPAS-Ocean" platform are carried out to numerically demonstrate the performance and parallel scalability of the proposed SSPRK-SE schemes.

math.NA↗

arXiv2021

Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Canonical Hamiltonian Systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of canonical Hamiltonian systems. Traditional intrusive projection-based model reduction approaches utilize symplectic Galerkin projection to construct Hamiltonian ROMs by projecting Hamilton's equations of the full model onto a symplectic subspace. This symplectic projection requires complete knowledge about the full model operators and full access to manipulate the computer code. In contrast, the proposed Hamiltonian operator inference approach embeds the physics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying symplectic structure. Our method exploits knowledge of the Hamiltonian functional to define and parametrize a Hamiltonian ROM form which can then be learned from data projected via symplectic projectors. The proposed method is gray-box in that it utilizes knowledge of the Hamiltonian structure at the partial differential equation level, as well as knowledge of spatially local components in the system. However, it does not require access to computer code, only data to learn the models. Our numerical results demonstrate Hamiltonian operator inference on a linear wave equation, the cubic nonlinear Schrödinger equation, and a nonpolynomial sine-Gordon equation. Accurate long-time predictions far outside the training time interval for nonlinear examples illustrate the generalizability of our learned models.

math.NA↗

arXiv2021

Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), in this paper we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

cs.LG↗

arXiv2021

A hyper-reduced MAC scheme for the parametric Stokes and Navier-Stokes equations

The need for accelerating the repeated solving of certain parametrized systems motivates the development of more efficient reduced order methods. The classical reduced basis method is popular due to an offline-online decomposition and a mathematically rigorous {\em a posterior} error estimator which guides a greedy algorithm offline. For nonlinear and nonaffine problems, hyper reduction techniques have been introduced to make this decomposition efficient. However, they may be tricky to implement and often degrade the online computation efficiency. To avoid this degradation, reduced residual reduced over-collocation (R2-ROC) was invented integrating empirical interpolation techniques on the solution snapshots and well-chosen residuals, the collocation philosophy, and the simplicity of evaluating the hyper-reduced well-chosen residuals. In this paper, we introduce an adaptive enrichment strategy for R2-ROC rendering it capable of handling parametric fluid flow problems. Built on top of an underlying Marker and Cell (MAC) scheme, a novel hyper-reduced MAC scheme is therefore presented and tested on Stokes and Navier-Stokes equations demonstrating its high efficiency, stability and accuracy.

math.NA↗

arXiv2021

An efficient iterative method for solving parameter-dependent and random convention-diffusion problems

This paper develops and analyzes a general iterative framework for solving parameter-dependent and random convection-diffusion problems. It is inspired by the multi-modes method of [7,8] and the ensemble method of [20] and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with parameter-independent convection and diffusion coefficients and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the $LU$ matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion and convection-diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.

math.NA↗

arXiv2021

Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations

A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.

stat.ML↗

arXiv2021

Bilinear Control of Convection-Cooling: From Open-Loop to Closed-Loop

This paper is concerned with a bilinear control problem for enhancing convection-cooling via an incompressible velocity field. Both optimal open-loop control and closed-loop feedback control designs are addressed. First and second order optimality conditions for characterizing the optimal solution are discussed. In particular, the method of instantaneous control is applied to establish the feedback laws. Moreover, the construction of feedback laws is also investigated by directly utilizing the optimality system with appropriate numerical discretization schemes. Computationally, it is much easier to implement the closed-loop feedback control than the optimal open-loop control, as the latter requires to solve the state equations forward in time, coupled with the adjoint equations backward in time together with a nonlinear optimality condition. Rigorous analysis and numerical experiments are presented to demonstrate our ideas and validate the efficacy of the control designs.

math.OC↗

arXiv2021

Parallel Exponential Time Differencing Methods for Geophysical Flow Simulations

Two ocean models are considered for geophysical flow simulations: the multi-layer shallow water equations and the multi-layer primitive equations. For the former, we investigate the parallel performance of exponential time differencing (ETD) methods, including exponential Rosenbrock-Euler, ETD2wave, and B-ETD2wave. For the latter, we take advantage of the splitting of barotropic and baroclinic modes and propose a new two-level method in which an ETD method is applied to solve the fast barotropic mode. These methods could improve the computational efficiency of numerical simulations because ETD methods allow for much larger time step sizes than traditional explicit time-stepping techniques that are commonly used in existing computational ocean models. Several standard benchmark tests for ocean modeling are performed and comparison of the numerical results demonstrate a great potential of applying the parallel ETD methods for simulating real-world geophysical flows.

physics.ao-ph↗

arXiv2021

A FOM/ROM Hybrid Approach for Accelerating Numerical Simulations

The basis generation in reduced order modeling usually requires multiple high-fidelity large-scale simulations that could take a huge computational cost. In order to accelerate these numerical simulations, we introduce a FOM/ROM hybrid approach in this paper. It is developed based on an a posteriori error estimation for the output approximation of the dynamical system. By controlling the estimated error, the method dynamically switches between the full-order model and the reduced-oder model generated on the fly. Therefore, it reduces the computational cost of a high-fidelity simulation while achieving a prescribed accuracy level. Numerical tests on the non-parametric and parametric PDEs illustrate the efficacy of the proposed approach.

math.NA↗