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First-principles determination of the phonon-point defect scattering and thermal transport due to fission products in ThO2

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Physical Review Materials 2024年第2期8卷 025401-025401页

作者： Linu Malakkal Ankita Katre Shuxiang Zhou Chao Jiang David H. Hurley Chris A. Marianetti Marat Khafizov Computational Mechanics and Materials Department Idaho National Laboratory Idaho Falls Idaho 83415 USA Department of Scientific Computing Modeling and Simulation SP Pune University Pune 411007 India Idaho National Laboratory Idaho Falls Idaho 83415 USA Department of Applied Physics and Applied Mathematics Columbia University 500 West 120th Street New York New York 10027 USA Department of Mechanical and Aerospace Engineering The Ohio State University 201 West 19th Avenue Columbus Ohio 43210 USA

This work presents the first-principles calculations of the lattice thermal conductivity degradation due to point defects in thorium dioxide using an iterative solution of the Peierls-Boltzmann transport equation. We have used the nonperturbative Green's function methodology to compute the phonon-point defect scattering rates that consider the local distortion around the point defect, including the mass difference changes, interatomic force constants, and structural relaxation near the point defects. The point defects considered in this work include the vacancy of thorium (VTh) and oxygen (VO), substitutions of helium (HeTh), krypton (KrTh), zirconium (ZrTh), iodine (ITh), and xenon (XeTh) in the thorium site, and the three different configurations of the Schottky defects. The results of the phonon-defect scattering rate reveal that among all the considered intrinsic defects, the thorium vacancy and helium substitution in the thorium site scatter the phonon most due to the substantial changes in the force constant and structural distortions. The scattering of phonons due to the substitutional defects unveils that the zirconium atom scatters phonons the least, followed by xenon, iodine, krypton, and helium. This is contrary to the intuition that the scattering strength follows HeTh>KrTh>ZrTh>ITh>XeTh based on the mass difference. This striking difference in the zirconium phonon scattering is due to the local chemical environment changes. Zirconium is an electropositive element with valency similar to thorium and, therefore, can bond with the oxygen atoms, thus creating less force constant variance compared to iodine, an electronegative element, and the noble gases helium, xenon, and krypton. These results can serve as a benchmark for analytical models and help the engineering-scale modeling effort for nuclear design.

关键词： Carrier dynamics Energy materials Heat transfer Irradiation effects Lattice dynamics Nuclear power Phonons Thermal conductivity Thermal properties

The positive mass theorem for asymptotically flat manifolds with isolated conical singularities Dedicated to Professor Weiping Zhang on the Occasion of His 60th Birthday

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Science China mathematics 2025年第7期 1671-1686页

作者： Xianzhe Dai Yukai Sun Changliang Wang Department of Mathematics University of CaliforniaSanta Barbara Key Laboratory of Pure and Applied Mathematics School of Mathematical SciencesPeking University School of Mathematical Sciences and Institute for Advanced Study Key Laboratory of Intelligent Computing and Applications (Ministry of Education)Tongji University

We prove the positive mass theorem for asymptotically flat(AF for short) manifolds with finitely many isolated conical singularities. We do not impose the spin condition. Instead, we use the conformal blow-up techniqu...

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Tensor neural networks for high-dimensional Fokker-Planck equations

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arXiv 2024年

作者： Wang, Taorui Hu, Zheyuan Kawaguchi, Kenji Zhang, Zhongqiang Karniadakis, George Em Department of Mathematical Sciences Worcester Polytechnic Institute WorcesterMA United States Department of Computer Science National University of Singapore 119077 Singapore Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor products of one-dimensional feedforward networks or a linear combination of several selected radial basis functions. The use of tensor feedforward networks allows us to efficiently exploit auto-differentiation (in physical variables) in major Python packages while using radial basis functions can fully avoid auto-differentiation, which is rather expensive in high dimensions. We then use the physics-informed neural networks and stochastic gradient descent methods to learn the tensor networks. One essential step is to determine a proper bounded domain or numerical support for the Fokker-Planck equation. To better train the tensor radial basis function networks, we impose some constraints on parameters, which lead to relatively high accuracy. We demonstrate numerically that the tensor neural networks in physics-informed machine learning are efficient for steady-state Fokker-Planck equations from two to ten dimensions. Copyright © 2024, The Authors. All rights reserved.

关键词： Radial basis function networks

Unusual thermal properties of graphene origami crease:A molecular dynamics study

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Green Energy & Environment 2022年第1期7卷 86-94页

作者： Ning Wei Yang Chen Kun Cai Yingyan Zhang Qingxiang Pei Jin-Cheng Zheng Yiu-Wing Mai Junhua Zhao Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology Jiangnan UniversityWuxi214122China Shanghai Institute of Applied Mathematics and Mechanics Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghai200072China School of Engineering RMIT UniversityBundooraVIC3083Australia Institute of High Performance Computing A^(*)STAR138632Singapore Department of Physics and the Collaborative Innovation Center for Optoelectronic Semiconductors and Efficient Devices Xiamen UniversityXiamen361005China Xiamen University Malaysia 439000SepangSelangorMalaysia Centre for Advanced Materials Technology(CAMT) School of AerospaceMechanical and Mechatronic Engineering J07The University of SydneySydneyNSW2006Australia

Graphene is a two-dimensional material that can be folded into diverse and yet interesting nanostructures like macro-scale paper *** of graphene not only makes different morphological configurations but also modifies their mechanical and thermal *** by paper origami,herein we studied systemically the effects of creases,where sp^(2)to sp^(3)bond transformation occurs,on the thermal properties of graphene origami using molecular dynamics(MD)*** MD simulation results show that tensile strain reduces(not increases)the interfacial thermal resistance owing to the presence of the *** unusual phenomenon is explained by the micro-heat flux migration and stress *** findings on the graphene origami enable the design of the next-generation thermal management devices and flexible electronics with tuneable properties.

关键词： Graphene Origami Bond transformation Interfacial thermal resistance Molecular dynamics

Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

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arXiv 2023年

作者： Hu, Zheyuan Yang, Zhouhao Wang, Yezhen Karniadakis, George Em Kawaguchi, Kenji Division of Applied Mathematics Brown University ProvidenceRI02912 United States Department of Computer Science National University of Singapore Singapore119077 Singapore Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine

关键词： Stochastic systems

Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks

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arXiv 2023年

作者： Hu, Zheyuan Shi, Zekun Karniadakis, George Em Kawaguchi, Kenji Department of Computer Science National University of Singapore Singapore119077 Singapore Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss function calculation. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing the Hutchinson Trace Estimation (HTE) method. Starting with the second-order high-dimensional PDEs, which are ubiquitous in scientific computing, HTE is applied to transform the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach not only alleviates the computational bottleneck via Taylor-mode automatic differentiation but also significantly reduces memory consumption from the Hessian matrix to an HVP's scalar output. We further showcase HTE's convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with the Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variability and variance among dimensions. We further extend the application of HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackl

关键词： Stochastic systems

Rethinking Skip Connections in Spiking Neural Networks with Time-To-First-Spike Coding

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arXiv 2023年

作者： Kim, Youngeun Kahana, Adar Yin, Ruokai Li, Yuhang Stinis, Panos Karniadakis, George Em Panda, Priyadarshini Department of Electrical Engineering Yale University New HavenCT United States Division of Applied Mathematics Brown University ProvidenceRI United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

Time-To-First-Spike (TTFS) coding in Spiking Neural Networks (SNNs) offers significant advantages in terms of energy efficiency, closely mimicking the behavior of biological neurons. In this work, we delve into the role of skip connections, a widely used concept in Artificial Neural Networks (ANNs), within the domain of SNNs with TTFS coding. Our focus is on two distinct types of skip connection architectures: (1) addition-based skip connections, and (2) concatenation-based skip connections. We find that addition-based skip connections introduce an additional delay in terms of spike timing. On the other hand, concatenation-based skip connections circumvent this delay but produce time gaps between after-convolution and skip connection paths, thereby restricting the effective mixing of information from these two paths. To mitigate these issues, we propose a novel approach involving a learnable delay for skip connections in the concatenation-based skip connection architecture. This approach successfully bridges the time gap between the convolutional and skip branches, facilitating improved information mixing. We conduct experiments on public datasets including MNIST and Fashion-MNIST, illustrating the advantage of the skip connection in TTFS coding architectures. Additionally, we demonstrate the applicability of TTFS coding on beyond image recognition tasks and extend it to scientific machine-learning tasks, broadening the potential uses of SNNs. © 2023, CC BY.

关键词： Image recognition

Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks

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arXiv 2024年

作者： Hu, Zheyuan Kawaguchi, Kenji Zhang, Zhongqiang Em Karniadakis, George Department of Computer Science National University of Singapore 119077 Singapore Department of Mathematical Sciences Worcester Polytechnic Institute WorcesterMA01609 United States Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence;in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions. © 2024, CC BY.

关键词： Monte Carlo methods

Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Lévy Equations

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arXiv 2024年

作者： Hu, Zheyuan Zhang, Zhongqiang Karniadakis, George Em Kawaguchi, Kenji Department of Computer Science National University of Singapore Singapore119077 Singapore Department of Mathematical Sciences Worcester Polytechnic Institute WorcesterMA01609 United States Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

We introduce an innovative approach for solving high-dimensional Fokker-Planck-Lévy (FPL) equations in modeling non-Brownian processes across disciplines such as physics, finance, and ecology. We utilize a fractional score function and Physical-informed neural networks (PINN) to lift the curse of dimensionality (CoD) and alleviate numerical overflow from exponentially decaying solutions with dimensions. The introduction of a fractional score function allows us to transform the FPL equation into a second-order partial differential equation without fractional Laplacian and thus can be readily solved with standard physics-informed neural networks (PINNs). We propose two methods to obtain a fractional score function: fractional score matching (FSM) and score-fPINN for fitting the fractional score function. While FSM is more cost-effective, it relies on known conditional distributions. On the other hand, score-fPINN is independent of specific stochastic differential equations (SDEs) but requires evaluating the PINN model's derivatives, which may be more costly. We conduct our experiments on various SDEs and demonstrate numerical stability and effectiveness of our method in dealing with high-dimensional problems, marking a significant advancement in addressing the CoD in FPL equations. © 2024, CC BY.

关键词： Cost effectiveness