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arXiv 2020年

作者： Hu, Yihao Zhao, Tong Xu, Shixin Lin, Lizhen Xu, Zhiliang Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46545 United States Department of Computer Science and Engineering University of Notre Dame Notre DameIN46545 United States Duke Kunshan University Jiangsu Kunshan215316 China

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is frequently a challenging task. Inspired by the traditional finite difference and finite elements methods and emerging advancements in machine learning, we propose a sequence-to-sequence learning (Seq2Seq) framework called Neural-PDE, which allows one to automatically learn governing rules of any time-dependent PDE system from existing data by using a bidirectional LSTM encoder, and predict the solutions in next n time steps. One critical feature of our proposed framework is that the Neural-PDE is able to simultaneously learn and simulate all variables of interest in a PDE system. We test the Neural-PDE by a range of examples, from one-dimensional PDEs to a multi-dimensional and nonlinear complex fluids model. The results show that the Neural-PDE is capable of learning the initial conditions, boundary conditions and differential operators defining the initial-boundary-value problem of a PDE system without the knowledge of the specific form of the PDE system. In our experiments, the Neural-PDE can efficiently extract the dynamics within 20 epochs training and produce accurate predictions. Furthermore, unlike the traditional machine learning approaches for learning PDEs, such as CNN and MLP, which require great quantity of parameters for model precision, the Neural-PDE shares parameters among all time steps, and thus considerably reduces computational complexity and leads to a fast learning algorithm. Copyright © 2020, The Authors. All rights reserved.

关键词： Complex networks

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A Lagrangian Method for Reactive Transport with Solid/Aqueous Chemical Phase Interaction i

arXiv

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arXiv 2018年

作者： Schmidt, Michael J. Pankavich, Stephen D. Navarre-Sitchler, Alexis Benson, David A. Hydrologic Science and Engineering Program Department of Geology and Geological Engineering Colorado School of Mines GoldenCO80401 United States Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO80401 United States

A significant drawback of Lagrangian (particle-tracking) reactive transport models has been their inability to properly simulate interactions between solid and liquid chemical phases, such as dissolution and precipitation reactions. This work addresses that problem by implementing a mass-transfer algorithm between mobile and immobile sets of particles that allows aqueous species of reactant that are undergoing transport to interact with stationary solid species. This mass-transfer algorithm is demonstrated to solve the diffusion equation for an arbitrarily small level of diffusion and thus does not introduce any spurious mixing. The algorithm can be combined with random walks to simulate the desired total level of diffusion in a reactive transport system. Copyright © 2018, The Authors. All rights reserved.

关键词： Diffusion

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hIPPYlib-MUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertainty

arXiv

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arXiv 2021年

作者： Kim, Ki-Tae Villa, Umberto Parno, Matthew Marzouk, Youssef Ghattas, Omar Petra, Noemi University of California Merced Applied Mathematics School of Natural Sciences MercedCA United States The University of Texas at Austin Oden Institute for Computational Engineering & Sciences AustinTX United States Dartmouth College Department of Mathematics HanoverNH United States Massachusetts Institute of Technology Department of Aeronautics and Astronautics CambridgeMA United States The University of Texas at Austin Oden Institute for Computational Engineering & Sciences Department of Mechanical Engineering Department of Geological Sciences AustinTX United States

Bayesian inference provides a systematic framework for integration of data with mathematical models to quantify the uncertainty in the solution of the inverse problem. However, the solution of Bayesian inverse problems governed by complex forward models described by partial differential equations (PDEs) remains prohibitive with black-box Markov chain Monte Carlo (MCMC) methods. We present hIPPYlib-MUQ, an extensible and scalable software framework that contains implementations of state-of-the art algorithms aimed to overcome the challenges of high-dimensional, PDE-constrained Bayesian inverse problems. These algorithms accelerate MCMC sampling by exploiting the geometry and intrinsic low-dimensionality of parameter space via derivative information and low rank approximation. The software integrates two complementary open-source software packages, hIPPYlib and MUQ. hIPPYlib solves PDE-constrained inverse problems using automatically-generated adjoint-based derivatives, but it lacks full Bayesian capabilities. MUQ provides a spectrum of powerful Bayesian inversion models and algorithms, but expects forward models to come equipped with gradients and Hessians to permit large-scale solution. By combining these two complementary libraries, we created a robust, scalable, and efficient software framework that realizes the benefits of each and allows us to tackle complex large-scale Bayesian inverse problems across a broad spectrum of scientific and engineering disciplines. To illustrate the capabilities of hIPPYlib-MUQ, we present a comparison of a number of MCMC methods available in the integrated software on several high-dimensional Bayesian inverse problems. These include problems characterized by both linear and nonlinear PDEs, various noise models, and different parameter dimensions. The results demonstrate that large (∼ 50×) speedups over conventional black box and gradient-based MCMC algorithms can be obtained by exploiting Hessian information (from the log-posterior

关键词： Inverse problems

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Erratum to: An efficient forward model of the climate controls on interannual variation in tree-ring width

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Climate Dynamics 2011年第11期36卷 2441-2445页

作者： Susan E. Tolwinski-Ward Michael N. Evans Malcolm K. Hughes Kevin J. Anchukaitis Program in Applied Mathematics University of Arizona Tucson USA Department of Geology University of Maryland College Park USA Laboratory of Tree Ring Research University of Arizona Tucson USA Lamont-Doherty Earth Observatory Columbia University Palisades USA

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An Optimal Transport Approach for Network Regression

An Optimal Transport Approach for Network Regression

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Control Technology and Applications (CCTA),

作者： Alex G. Zalles Kai M. Hung Ann E. Finneran Lydia Beaudrot César A. Uribe Department of Computational Applied Mathematics and Operations Research Rice University Houston TX USA Department of Computer Science Rice University Houston TX USA Department of Biosciences Rice University Houston TX USA Department of Electrical and Computer Engineering Rice University Houston TX USA

ISBN: (数字)9798350370942

ISBN: (纸本)9798350370959

We study the problem of network regression, where one is interested in how the topology of a network changes as a function of Euclidean covariates. We build upon recent developments in generalized regression models on metric spaces based on Fréchet means and propose a network regression method using the Wasserstein metric. We show that when representing graphs as multivariate Gaussian distributions, the network regression problem requires the computation of a Riemannian center of mass (i.e., Fréchet means). Fréchet means with non-negative weights translates into a barycenter problem and can be efficiently computed using fixed point iterations. Although the convergence guarantees of fixed-point iterations for the computation of Wasserstein affine averages remain an open problem, we provide evidence of convergence in a large number of synthetic and real-data scenarios. Extensive numerical results show that the proposed approach improves existing procedures by accurately accounting for graph size, topology, and sparsity in synthetic experiments. Additionally, real-world experiments using the proposed approach result in higher Coefficient of Determination (R 2 ) values and lower mean squared prediction error (MSPE), cementing improved prediction capabilities in practice.

关键词： Accuracy Network topology computational modeling Cement industry Predictive models Gaussian distribution Aerospace electronics

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Note on the gravitational and electromagnetic radiation for the Einstein-Maxwell equations with cosmological constant

arXiv

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arXiv 2017年

作者： He, Xiaokai Cao, Zhoujian Jing, Jiliang School of Mathematics and Computational Science Hunan First Normal University Changsha Hunan410205 China Department of Physics Hunan Normal University Changsha Hunan410081 China Institute of Applied Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Physics Key Laboratory of Low Dimensional Quantum Structures Quantum Control of Ministry of Education Synergetic Innovation Center for Quantum Effects and Applications Hunan Normal University Changsha Hunan410081 China

In the middle of last century, Bondi and his coworkers proposed an out going boundary condition for the Einstein equations. Based on such boundary condition the authors theoretically solved the puzzle of the existence problem of gravitational wave. Currently many works on gravitational wave source modeling use Bondi's result to do the analysis including the gravitational wave calculation in numerical relativity. Recently more and more observations imply that the Einstein equations should be modified with an nonzero cosmological constant. In this work we use Bondi's original method to treat the Einstein equations with cosmological constant for theoretical curiosity. We find that the gravitational wave does not essentially exist if Bondi's original out going boundary condition is imposed. We find this unphysical result is due to the non-consistency between Bondi's original out going boundary condition and the Einstein equations with cosmological constant. Based on theoretical analysis, we propose an new Bondi-type out going boundary condition for the Einstein equations with cosmological constant. With this new boundary condition, the gravitational wave behavior for the Einstein equations with cosmological constant is similar to the Einstein equations without cosmological constant. In particular, when the cosmological constant goes to zero, the behavior of the Einstein equations without cosmological constant is recovered. Together with gravitational wave, electromagnetic wave is also analyzed. The boundary conditions for electromagnetic wave does not depend on the cosmological constant. Copyright © 2017, The Authors. All rights reserved.

关键词： Boundary conditions

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Original Article: A New Method for Optimal Solution of Intuitionistic Fuzzy Transportation Problems via Generalized Trapezoidal Intuitionistic Fuzzy Numbers

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Fuzzy Information and Engineering 2024年第1期11卷 105-120页

作者： Darunee Hunwisai Poom Kumam Wiyada Kumam Department of Applied Mathematics Faculty of Science and Technology Valaya Alongkorn Rajabhat University under the Royal Patronage Pathumthani Thailand Science Laboratory Building Faculty of Science KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA) Theoretical and Computational Science Center (TaCS) King Mongkut's University of Technology Thonburi (KMUTT) Bangkok Thailand Department of Mathematics and Computer Science Faculty of Science and Technology Rajamangala University of Technology Thanyaburi Pathumthani Thailand

In this paper, we introduce the new method for solving the intuitionistic fuzzy transportation problem (IFTP), by using north-west corner method and modified distribution method to find the optimal solution for IFTP.

关键词： Costs Shape Transportation

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Analysis of hyperspectral data with diffusion maps and Fuzzy ART

Analysis of hyperspectral data with diffusion maps and Fuzzy...

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International Joint Conference on Neural Networks (IJCNN)

作者： Rui Xu Louis du Plessis Steven Damelin Michael Sears Donald C. Wunsch Applied Computational Intelligence Laboratory Department of Electrical & Computer Engineering Missouri University of Science and Technology MO USA School of Computational and Applied Mathematics University of the Witwatersrand Johannesburg South Africa Department of Mathematical Sciences Georgia Southern University Statesboro GA USA School of Computer Science University of the Witwatersrand Johannesburg South Africa Department of Electrical & Computer Engineering Missouri University of Science and Technology Rolla MO USA

The presence of large amounts of data in hyperspectral images makes it very difficult to perform further tractable analyses. Here, we present a method of analyzing real hyperspectral data by dimensionality reduction using diffusion maps. Diffusion maps interpret the eigenfunctions of Markov matrices as a system of coordinates on the original data set in order to obtain an efficient representation of data geometric descriptions. A neural network clustering theory, Fuzzy ART, is further applied to the reduced data to form clusters of the potential minerals. Experimental results on a subset of hyperspectral core imager data show that the proposed methods are promising in addressing the complicated hyperspectral data and identifying the minerals in core samples.

关键词： Data analysis Hyperspectral imaging Subspace constraints Frequency Minerals Optical imaging Multispectral imaging Pixel Africa Neural networks

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Transferable Neural Wavefunctions for Solids

arXiv

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arXiv 2024年

作者： Gerard, Leon Scherbela, Michael Sutterud, Halvard Foulkes, W.C. Matthew Grohs, Philipp Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 ViennaA-1090 Austria Department of Physics Imperial College London South Kensington Campus LondonSW7 2AZ United Kingdom Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstrasse 69 Linz4040 Austria

Deep-Learning-based Variational Monte Carlo (DL-VMC) has recently emerged as a highly accurate approach for finding approximate solutions to the many-electron Schrödinger equation. Despite its favorable scaling with the number of electrons, O(nel4), the practical value of DL-VMC is limited by the high cost of optimizing the neural network weights for every system studied. To mitigate this problem, recent research has proposed optimizing a single neural network across multiple systems, reducing the cost per system. Here we extend this approach to solids, where similar but distinct calculations using different geometries, boundary conditions, and supercell sizes are often required. We show how to optimize a single ansatz across all of these variations, reducing the required number of optimization steps by an order of magnitude. Furthermore, we exploit the transfer capabilities of a pre-trained network. We successfully transfer a network, pre-trained on 2 × 2 × 2 supercells of LiH, to 3 × 3 × 3 supercells. This reduces the number of optimization steps required to simulate the large system by a factor of 50 compared to previous work. Copyright © 2024, The Authors. All rights reserved.

关键词： Deep learning

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Coarse-graining conformational dynamics with multi-dimensional generalized Langevin equation: how, when, and why

arXiv

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arXiv 2024年

作者： Xie, Pinchen Qiu, Yunrui Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry University of Wisconsin-Madison MadisonWI53706 United States AI for Science Institute Beijing100080 China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing100084 China

A data-driven ab initio generalized Langevin equation (AIGLE) approach is developed to learn and simulate high-dimensional, heterogeneous, coarse-grained conformational dynamics. Constrained by the fluctuation-dissipation theorem, the approach can build coarse-grained models in dynamical consistency with all-atom molecular dynamics. We also propose practical criteria for AIGLE to enforce long-term dynamical consistency. Case studies of a toy polymer, with 20 coarse-grained sites, and the alanine dipeptide, with two dihedral angles, elucidate why one should adopt AIGLE or its Markovian limit for modeling coarse-grained conformational dynamics in practice. © 2024, CC BY.

关键词： Coarse-grained modeling

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