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PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units

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数学研究 2020年第2期53卷 159-191页

作者： Bo Li Shanshan Tang Haijun Yu NCMIS&LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing 100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China China Justice Big Data Institute Beijing 100043 China

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations only when smoothness is not required. In this paper, we construct deep neural networks with rectified power units (RePU), which can give better approximations for smooth functions. Optimal algorithms are proposed to explicitly build neural networks with sparsely connected RePUs, which we call PowerNets, to represent polynomials with no approximation error. For general smooth functions, we first project the function to their polynomial approximations, then use the proposed algorithms to construct cor-responding PowerNets. Thus, the error of best polynomial approximation provides an upper bound of the best RePU network approximation error. For smooth functions in higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product grid and sparse grid discretization to get polynomial approximations. Our construc-tive algorithms show clearly a close connection between spectral methods and deep neural networks: PowerNets with n hidden layers can exactly represent polynomials up to degree sn, where s is the power of RePUs. The proposed PowerNets have po-tential applications in the situations where high-accuracy is desired or smoothness is required.

关键词： Deep neural network rectified linear unit rectified power unit sparse grid Power-Net

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Symmetrized two-scale finite element discretizations for partial differential equations with symmetric solutions

arXiv

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arXiv 2022年

作者： Hou, Pengyu Liu, Fang Zhou, Aihui School of Mathematical Sciences Beijing Normal University Beijing100875 China School of Statistics and Mathematics Central University of Finance and Economics Beijing102206 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this paper, a symmetrized two-scale finite element method is proposed for a class of partial differential equations with symmetric solutions. With this method, the finite element approximation on a fine tensor product grid is reduced to the finite element approximations on a much coarse grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximation still maintains an asymptotically optimal accuracy. Consequently the symmetrized two-scale finite element method reduces computational cost *** Codes 65N15, 65N25, 65N30, 65N50 Copyright © 2022, The Authors. All rights reserved.

关键词： Partial differential equations

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A lowest-degree strictly conservative finite element scheme for incompressible stokes problem on general triangulations

arXiv

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arXiv 2021年

作者： Liu, Wenjia Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, we propose a finite element pair for incompressible Stokes problem. The pair uses a slightly enriched piecewise linear polynomial space for velocity and piecewise constant space for pressure, and is illustrated to be a lowest-degree conservative stable pair for the Stokes problem on general triangulations. Copyright © 2021, The Authors. All rights reserved.

关键词： Navier Stokes equations

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Minimum tracking linear response Hubbard and Hund corrected Density Functional Theory in CP2K

arXiv

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arXiv 2024年

作者： Chai, Ziwei Si, Rutong Chen, Mingyang Teobaldi, Gilberto O'Regan, David D. Liu, Li-Min Beijing Computational Science Research Center Beijing100193 China School of Physics Beihang University Beijing100191 China School of Materials Science and Engineering University of Science and Technology Beijing Beijing100083 China Scientific Computing Department STFC UKRI Rutherford Appleton Laboratory Harwell Campus DidcotOX11 0QX United Kingdom School of Physics SFI AMBER Centre CRANN Institute Trinity College Dublin The University of Dublin Ireland

We present the implementation of the Hubbard (U) and Hund (J) corrected Density Functional Theory (DFT+U+J) functionality in the Quickstep program, which is part of the CP2K suite. The tensorial and Löwdin subspace representations are implemented and compared. Full analytical DFT+U+J forces are implemented and benchmarked for the tensorial and Löwdin representations. We also present the implementation of the recently proposed minimum-tracking linear-response method that enables the U and J parameters to be calculated on first principles basis without reference to the Kohn-Sham eigensystem. These implementations are benchmarked against recent results for different materials properties including DFT+U band gap opening in NiO, the relative stability of various polaron distributions in TiO2, the dependence of the calculated TiO2 band gap on +J corrections, and, finally, the role of the +U and +J corrections for the computed properties of a series of the hexahydrated transition metals. Our implementation provides results consistent with those already reported in the literature from comparable methods. We conclude the contribution with tests on the influence of the Löwdin orthonormalization on the occupancies, calculated parameters, and derived properties. © 2024, CC BY.

关键词： Titanium dioxide

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Optimality conditions for homogeneous polynomial optimization on the unit sphere

arXiv

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arXiv 2021年

作者： Huang, Lei Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing China

In this note, we prove that for homogeneous polynomial optimization on the sphere, if the objective f is generic in the input space, all feasible points satisfying the first order and second order necessary optimality conditions are local minimizers, which addresses an issue raised in the recent work by Lasserre (Optimization Letters, 2021). As a corollary, this implies that Lasserre’s hierarchy has finite convergence when f is generic. Copyright © 2021, The Authors. All rights reserved.

关键词： Spheres

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MatterChat: A Multi-Modal LLM for Material Science

arXiv

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arXiv 2025年

作者： Tang, Yingheng Xu, Wenbin Cao, Jie Gao, Weilu Farrell, Steven Erichson, Benjamin Mahoney, Michael W. Nonaka, Andy Yao, Zhi Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA United States National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory BerkeleyCA United States NSF National AI Institute for Student-AI Teaming University of Colorado at Boulder Boulder United States Department of Electrical and Computer Engineering The University of Utah Salt Lake CityUT United States Scientific Data Division Lawrence Berkeley National Laboratory BerkeleyCA United States International Computer Science Institute BerkeleyCA United States Department of Statistics University of California at Berkeley BerkeleyCA United States

Understanding and predicting the properties of inorganic materials is crucial for accelerating advancements in materials science and driving applications in energy, electronics, and beyond. Integrating material structure data with language-based information through multi-modal large language models (LLMs) offers great potential to support these efforts by enhancing human–AI interaction. However, a key challenge lies in integrating atomic structures at full resolution into LLMs. In this work, we introduce MatterChat, a versatile structure-aware multi-modal LLM that unifies material structural data and textual inputs into a single cohesive model. MatterChat employs a bridging module to effectively align a pretrained universal machine learning interatomic potential with a pretrained LLM, reducing training costs and enhancing flexibility. Our results demonstrate that MatterChat significantly improves performance in material property prediction and human-AI interaction, surpassing general-purpose LLMs such as GPT-4. We also demonstrate its usefulness in applications such as more advanced scientific reasoning and step-by-step material synthesis. © 2025, CC BY.

关键词： Structure (composition)

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A DECOUPLING TWO-GRID METHOD FOR THE STEADY-STATE POISSON-NERNST-PLANCK EQUATIONS

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Journal of computational mathematics 2019年第4期37卷 556-578页

作者： Ying Yang Benzhuo Lu Yan Xie School of Mathematics and Computing Science Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation Guilin University of Electronic Technology Guilin 541004 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing the National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. Two kinds of two-grid finite element algorithms are proposed to decouple the steady-state Poisson-Nernst-Planck equations by coarse grid finite element approximations. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid algorithms for solving Poisson-Nernst-Planck equations.

关键词： Poisson-Nernst-Planck equations Two-grid finite element method Decoupling method Error analysis Gummel iteration

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Machine Learning With Data Assimilation and Uncertainty Quantification for Dynamical Systems:A Review

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IEEE/CAA Journal of Automatica Sinica 2023年第6期10卷 1361-1387页

作者： Sibo Cheng César Quilodrán-Casas Said Ouala Alban Farchi Che Liu Pierre Tandeo Ronan Fablet Didier Lucor Bertrand Iooss Julien Brajard Dunhui Xiao Tijana Janjic Weiping Ding Yike Guo Alberto Carrassi Marc Bocquet Rossella Arcucci Data Science Institute Department of ComputingImperial College LondonSW72AZ London Department of Earth Science and Engineering Imperial College LondonSW72AZ London Department of Computer Science and Engineering Hong Kong University of Science and TechnologyHong Kong 999077China the IMT Atlantique Lab-STICCUMR CNRS 6285France and OdysseyInria/IMTFrance.P.Tandeo is also with RIKEN Center for Computational ScienceKobeJapan the CEREA École des Ponts and EDF R&Dîle-de-FranceFrance the Laboratoire Interdisciplinaire des Sciences du Numérique CNRSParis-Saclay UniversityF-91403OrsayFrance the Electricitéde France(EDF) 78401 ChatouFranceInstitut de Mathématiques de Toulouse31062 ToulouseFrance and SINCLAIR AI LabSaclayFrance the Sorbonne University ParisFranceand also with Nansen Environmental and Remote Sensing Center(NERSC)BergenNorway the School of Mathematical Sciences Tongji UniversityShanghai 200092China the Mathematical Institute for Machine Learning and Data Science KU Eichstaett-IngolstadtBavariaGermany the School of Information Science and Technology Nantong UniversityNantong 226019China the Department of Physics and Astronomy“Augusto Righi” University of Bologna40124 BolognaItaly

Data assimilation(DA)and uncertainty quantification(UQ)are extensively used in analysing and reducing error propagation in high-dimensional spatial-temporal *** applications span from computational fluid dynamics(CFD)to geoscience and climate ***,much effort has been given in combining DA,UQ and machine learning(ML)*** research efforts seek to address some critical challenges in high-dimensional dynamical systems,including but not limited to dynamical system identification,reduced order surrogate modelling,error covariance specification and model error correction.A large number of developed techniques and methodologies exhibit a broad applicability across numerous domains,resulting in the necessity for a comprehensive *** paper provides the first overview of state-of-the-art researches in this interdisciplinary field,covering a wide range of *** review is aimed at ML scientists who attempt to apply DA and UQ techniques to improve the accuracy and the interpretability of their models,but also at DA and UQ experts who intend to integrate cutting-edge ML approaches to their ***,this article has a special focus on how ML methods can overcome the existing limits of DA and UQ,and vice *** exciting perspectives of this rapidly developing research field are also *** Terms-Data assimilation(DA),deep learning,machine learning(ML),reduced-order-modelling,uncertainty quantification(UQ).

关键词： assimilation overcome apply

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HOMOGENIZATION FOR POLYNOMIAL OPTIMIZATION WITH UNBOUNDED SETS

arXiv

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arXiv 2021年

作者： Huang, Lei Nie, Jiawang Yuan, Ya-Xiang Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100190 China Department of Mathematics University of California 9500 Gilman Drive La Jolla San DiegoCA92093 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work. Polynomial optimization and Homogenization and Moment-SOS relaxations and Optimality conditions. Copyright © 2021, The Authors. All rights reserved.

关键词： Polynomials

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Efficient CI-Based One-Bit Precoding for Multiuser Downlink Massive MIMO Systems with PSK Modulation

arXiv

引用

arXiv 2021年

作者： Wu, Zheyu Jiang, Bo Liu, Ya-Feng Shao, Mingjie Dai, Yu-Hong The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China The Ministry of Education Key Laboratory for NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing210023 China The School of Information Science and Engineering Shandong University Qingdao China

In this paper, we consider the one-bit precoding problem for the multiuser downlink massive multiple-input multiple-output (MIMO) system with phase shift keying (PSK) modulation. We focus on the celebrated constructive interference (CI)-based problem formulation. We first establish the NP-hardness of the problem (even in the single-user case), which reveals the intrinsic difficulty of globally solving the problem. Then, we propose a novel negative 1 penalty model for the considered problem, which penalizes the one-bit constraint into the objective by a negative 1-norm term, and show the equivalence between (global and local) solutions of the original problem and the penalty problem when the penalty parameter is sufficiently large. We further transform the penalty model into an equivalent min-max problem and propose an efficient alternating proximal/projection gradient descent ascent (APGDA) algorithm for solving it, which performs a proximal gradient decent over one block of variables and a projection gradient ascent over the other block of variables alternately. The APGDA algorithm enjoys a low per-iteration complexity and is guaranteed to converge to a stationary point of the min-max problem and a local minimizer of the penalty problem. To further reduce the computational cost, we also propose a low-complexity implementation of the APGDA algorithm, where the values of the variables will be fixed in later iterations once they satisfy the one-bit constraint. Numerical results show that, compared to the state-of-the-art CI-based algorithms, both of the proposed algorithms generally achieve better bit-error-rate (BER) performance with lower computational cost. Copyright © 2021, The Authors. All rights reserved.

关键词： Gradient methods

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