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Multigoal-oriented dual-weighted-residual error estimation using PINNs

Machine Learning for computational Science and Engineering 2025年第1期1卷 1-18页

作者： Chakraborty, Ayan Wick, Thomas Rabczuk, Timon Zhuang, Xiaoying Chair of Computational Science and Simulation Technology Institute of Photonics Faculty of Mathematics and Physics Leibniz University Hannover Germany Institute of Applied Mathematics Welfengarten 1 AG Wissenschaftliches Rechnen Leibniz University Hannover Germany Cluster of Excellence PhoenixD (Photonics Optics and Engineering - Innovation Across Disciplines) Leibniz University Hannover Hannover Germany Université Paris-Saclay ENS Paris-Saclay LMT - Laboratoire de Mecanique et Technologie Gif-sur-Yvette France Institute of Structural Mechanics Bauhaus University Weimar Weimar Germany Department of Geotechnical Engineering College of Civil Engineering Tongji University Shanghai China

Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. It is mainly considered as a powerful tool with high flexibility to approximate functions. This paper proposes a generalized NURBS based approach to solve nonlinear partial differential equations (PDEs) on arbitrary complex-geometry domains by using physics-informed neural networks (PINNs). Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network. An efficient and easy to implement algorithm is developed to obtain a posteriori error estimate for multiple goal functionals by employing the dual-weighted residual approach, which is followed by the computation of both primal and adjoint solutions using the neural network. The present study shows that such a data-driven model based learning has superior approximation of quantities of interest even with relatively less training data. Moreover, we illustrate the versatility of activation functions in achieving better learning capabilities and improving convergence rates, especially at the early training stage, and also in increasing solutions accuracies. The novel algorithmic developments are substantiated with several numerical test examples. It has been demonstrated that deep neural networks have distinct advantages over shallow neural networks, and the techniques for enhancing convergence have also been reviewed.

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Design and Characterization of Type-Ii Superlattice-Based Inas/Alsb/Gasb Detector Structure

SSRN

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SSRN 2024年

作者： Kızılkaya, Kürşat Öztürk, Mustafa Kemal Hoştut, Mustafa Ergün, Yüksel Özçelik, Süleyman Turkish Aerospace Fethiye District Havacılık Avenue No:17 Ankara06980 Turkey Advanced Technologies Program Graduate School of Natural and Applied Sciences Gazi University Ankara06560 Turkey Photonics Application and Research Center Gazi University Ankara06560 Turkey Physics Department of Science Faculty Gazi University Ankara06560 Turkey Physics Department of Science Faculty Eskisehir Technical University Eskisehir26470 Turkey Photonics Department of Applied Science Faculty Gazi University Ankara06560 Turkey Department of Mathematics and Science Education Faculty of Education Akdeniz University Antalya07058 Turkey

In this study, N-Structured based InAs/AlSb/GaSb Type-II Superlattice pbin type detector structures are investigated. These systems make absorption in infrared region in electromagnetic spectrum as detectors. Structural characterization of these systems is aimed. n-type GaSb is used as the substrate. InAs/AlSb/GaSb-based repetitive SL layers are formed as n-layer, i-layer, p-layer and p-contact layer from the bottom InAsSb layer to the top GaSb cap layer for 120, 300, 15 and 80 periods, respectively. Structural characterization of all layers is made by using SIMS and XRD systems analyses. The structural depth profile of the SL layers has been presented comparatively through SIMS analysis. The interfacial transitions observed in the depth profile are consistent with the designed T2SL layers. By using XRD characterization, crystal quality parameters, lattice constants and periodicity and deviation of super lattice peak from substrate are determined. Such dislocation density and strain are measured AS 7.85x107cm-2 and 6.4x10-3 nm, respectively. SEM analysis is used to examine top layer morphology and cross-section of the structure. The cross-sectional measurements allowed the observation of interfacial transitions within the SL structure. In this study, super lattice with high crystal quality is achieved and results show that T2SL detector structures are succeeded. © 2024, The Authors. All rights reserved.

关键词： III-V semiconductors

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Spread complexity as classical dilaton solutions

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Physical Review D 2023年第2期108卷 025013-025013页

作者： Arghya Chattopadhyay Arpita Mitra Hendrik J. R. van Zyl Service de Physique de l’Univers Champs et Gravitation Université de Mons 20 Place du Parc 7000 Mons Belgium Department of Physics Pohang University of Science and Technology Pohang 37673 Korea Laboratory for Quantum Gravity & Strings Department of Mathematics & Applied Mathematics University of Cape Town Cape Town 7700 Rondebosch South Africa The National Institute for Theoretical and Computational Sciences Private Bag X1 Matieland 7600 Stellenbosch South Africa

We demonstrate a relation between Nielsen’s approach toward circuit complexity and Krylov complexity through a particular construction of quantum state space geometry. We start by associating Kähler structures on the full projective Hilbert space of low rank algebras. This geometric structure of the states in the Hilbert space ensures that every unitary transformation of the associated algebras leave the metric and the symplectic forms invariant. We further associate a classical matter free Jackiw-Teitelboim gravity model with these state manifolds and show that the dilaton can be interpreted as the quantum mechanical expectation values of the symmetry generators. On the other hand, we identify the dilaton with the spread complexity over a Krylov basis thereby proposing a geometric perspective connecting two different notions of complexity.

关键词： Dualities in field theory Quantum field theory (low energy) Quantum-to-classical transition Classical black holes Group theory

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Atomic stick-slip friction as a two-dimensional thermally activated process

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Physical Review B 2022年第16期105卷 165429-165429页

作者： Quanzhou Yao Jiawei Sun Xiaoying Zhuang Peter Wriggers Xi-Qiao Feng Qunyang Li Applied Mechanics Laboratory Department of Engineering Mechanics Tsinghua University Beijing 100084 China Chair of Computational Science and Simulation Technology Institute of Photonics Department of Mathematics and Physics Leibniz University Hannover 30167 Hannover Germany Institute for Continuum Mechanics Leibniz University Hannover Garbsen Germany State Key Laboratory of Tribology Tsinghua University Beijing 100084 China

Widely recognized as a thermally activated process, atomic stick-slip friction has been typically explained by Prandtl-Tomlinson model with thermal activation. Despite the limited success, theoretical predictions from the classic model are primarily based on a one-dimensional (1D) assumption, which is generally not compatible with real experiments that are two-dimensional (2D) in nature. In this letter, a theoretical model based on 2D transition state theory has been derived and confirmed to be able to capture the 2D slip kinetics in atomic-scale friction experiments on crystalline surface with a hexagonal energy landscape. Moreover, we propose a reduced scheme that enables extraction of intrinsic interfacial parameters from 2D experiments approximately using the traditional 1D model. The 2D model provides a theoretical tool for understanding the rich kinetics of atomic-scale friction or other phenomena involving higher dimensional transitions.

关键词： Friction Atomic force microscopy Thermal & statistical models

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Laboratory investigation of the jet deflection: collisionless to strong-collision

arXiv

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arXiv 2022年

作者： Lei, Zhu Zhao, Z.H. Li, X.Y. Li, L.X. Xie, Y. Yuan, W.Q. Zhu, B.Q. Zhu, J.Q. Zhu, S.P. He, X.T. Qiao, B. Center for Applied Physics and Technology HEDPS State Key Laboratory of Nuclear Physics and Technology School of Physics Peking University Beijing100871 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Department of Plasma Physics and Fusion Engineering CAS Key Laboratory of Geospace Environment University of Science and Technology of China Anhui Hefei230026 China National Laboratory on High Power Laser and Physics Chinese Academy of Sciences Shanghai201800 China

The interaction between a jet and its surrounding medium plays a crucial role in various astrophysical phenomena, notably in star formation and jet deflection. Despite its significance, the detailed processes and fine shock structures involved in this interaction remain elusive due to limitations in current astronomical facilities, such as the confusing HCO+ emission in the Herbig-Haro (HH) 110/270 system. Here, we investigate the scaled plasma dynamics of this interaction under different collision states using laser-driven experiments. The experimental results demonstrate the importance of collision in jet deflection, revealing a distinctive double-shock structure in collision cases and a double-emission-shell structure in cases of strong collisions. Both cases exhibit a notable deflection of plasma jets by 50◦ as a consequence of interaction with a high-density background. Optical images confirm that the impact point can be pushed into the inside part of the background medium, consistent with previous astrophysical models and simulations. In contrast, collisionless cases exhibit filament structures (the ion skin depth scale), yet there is an absence of discernible deflection in the direction of jet transmission. Furthermore, our findings have implications for understanding the shock structure in the HH 110/270 system. The significant HCO+ emission observed west of HH 110 may be attributed to the impact point being pushed into the inside part of the background. Copyright © 2022, The Authors. All rights reserved.

关键词： Astrophysics

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Algebraic Compression of Free Fermionic Quantum Circuits: Particle Creation, Arbitrary Lattices and Controlled Evolution

arXiv

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arXiv 2023年

作者： Kökcü, Efekan Camps, Daan Oftelie, Lindsay Bassman de Jong, Wibe A. Van Beeumen, Roel Kemper, A.F. Department of Physics North Carolina State University RaleighNC27695 United States National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States

Recently we developed a local and constructive algorithm based on Lie algebraic methods for compressing Trotterized evolution under Hamiltonians that can be mapped to free fermions [1, 2]. The compression algorithm yields a circuit which scales linearly in the number of qubits, is fixed depth for for arbitrarily long evolution times and is applicable to time dependent Hamiltonians, although is limited to simple nearest-neighbor spin interactions and fermionic hopping. In this work, we extend the algorithm to compress circuits simulating evolution with long-range spin interactions and fermionic hopping, thereby enabling embedding of arbitrary lattices onto a chain of qubits. Moreover, we show that controlled time evolution, as well as fermion creation and annihilation operators can also be compressed. We demonstrate our results by adiabatically preparing the ground state for a half-filled fermionic chain, and simulating a 4×4 tight binding model on ibmq washington. With these new developments, our results enable the simulation of a wider range of models of interest and the efficient compression of subcircuits. Copyright © 2023, The Authors. All rights reserved.

关键词： Ground state

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Coarse-grained spectral projection (CGSP): A deep learning-assisted approach to quantum unitary dynamics

arXiv

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arXiv 2020年

作者： Xie, Pinchen Weinan, E. Program in Applied and Computational Mathematics Princeton University NJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We propose the coarse-grained spectral projection method (CGSP), a deep learning-assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show CGSP can extract spectral components of many-body quantum states systematically with sophisticated neural network quantum ansatz. CGSP exploits fully the linear unitary nature of the quantum dynamics, and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on 1D XXZ models with periodic boundary condition are carried out to demonstrate the practicality of CGSP. Copyright © 2020, The Authors. All rights reserved.

关键词： Dynamics

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KOLMOGOROV WIDTH DECAY AND POOR APPROXIMATORS IN MACHINE LEARNING: SHALLOW NEURAL NETWORKS, RANDOM FEATURE MODELS AND NEURAL TANGENT KERNELS

arXiv

引用

arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544

We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor L2-approximators for the class of two-layer neural networks in high dimension, and that two-layer networks with small path norm are poor approximators for certain Lipschitz functions, also in the *** Codes 68T07, 41A30, 41A65, 46E15, 46E22 Copyright © 2020, The Authors. All rights reserved.

关键词： Banach spaces

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DeePKS-kit: A package for developing machine learning-based chemically accurate energy and density functional models

arXiv

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arXiv 2020年

作者： Chen, Yixiao Zhang, Linfeng Wang, Han Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Mathematics Princeton University PrincetonNJ United States

We introduce DeePKS-kit, an open-source software package for developing machine learning based energy and density functional models. DeePKS-kit is interfaced with PyTorch, an open-source machine learning library, and PySCF, an ab initio computational chemistry program that provides simple and customized tools for developing quantum chemistry codes. It supports the DeePHF and DeePKS methods. In addition to explaining the details in the methodology and the software, we also provide an example of developing a chemically accurate model for water clusters. Copyright © 2020, The Authors. All rights reserved.

关键词： Electronic structure

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Ground state energy functional with Hartree-Fock efficiency and chemical accuracy

arXiv

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arXiv 2020年

We introduce the Deep Post–Hartree–Fock (DeePHF) method, a machine learning-based scheme for constructing accurate and transferable models for the ground-state energy of electronic structure problems. DeePHF predicts the energy difference between results of highly accurate models such as the coupled cluster method and low accuracy models such as the the Hartree-Fock (HF) method, using the ground-state electronic orbitals as the input. It preserves all the symmetries of the original high accuracy model. The added computational cost is less than that of the reference HF or DFT and scales linearly with respect to system size. We examine the performance of DeePHF on organic molecular systems using publicly available datasets and obtain the state-of-art performance, particularly on large datasets. Copyright © 2020, The Authors. All rights reserved.

关键词： Molecular physics

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