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Analysis of the gradient descent algorithm for a deep neural network model with skip-connections

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arXiv 2019年

作者： Weinan, E. Ma, Chao Wang, Qingcan Wu, Lei Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

The behavior of the gradient descent (GD) algorithm is analyzed for a deep neural network model with skip-connections. It is proved that in the over-parametrized regime, for a suitable initialization, with high probability GD can find a global minimum exponentially fast. Generalization error estimates along the GD path are also established. As a consequence, it is shown that when the target function is in the reproducing kernel Hilbert space (RKHS) with a kernel defined by the initialization, there exist generalizable earlystopping solutions along the GD path. In addition, it is also shown that the GD path is uniformly close to the functions given by the related random feature model. Consequently, in this \implicit regularization" setting, the deep neural network model deteriorates to a random feature model. Our results hold for neural networks of any width larger than the input dimension. Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods

mathematics for cryo-electron microscopy

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Mathematics for cryo-electron microscopy

2018 International Congress of Mathematicians, ICM 2018

作者： Singer, Amit Department of Mathematics Program in Applied and Computational Mathematic Princeton University United States

ISBN: (纸本)9789813272934

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM. © ICM *** rights reserved.

关键词： X ray crystallography

Hyperuniformity order metric of Barlow packings

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Physical Review E 2019年第2期99卷 022111-022111页

作者： T. M. Middlemas F. H. Stillinger S. Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University New Jersey 08544 USA

The concept of hyperuniformity has been a useful tool in the study of density fluctuations at large length scales in systems ranging across the natural and mathematical sciences. One can rank a large class of hyperuniform systems by their ability to suppress long-range density fluctuations through the use of a hyperuniformity order metric Λ¯. We apply this order metric to the Barlow packings, which are the infinitely degenerate densest packings of identical rigid spheres that are distinguished by their stacking geometries and include the commonly known fcc lattice and hcp crystal. The “stealthy stacking” theorem implies that these packings are all stealthy hyperuniform, a strong type of hyperuniformity, which involves the suppression of scattering up to a wave vector K. We describe the geometry of three classes of Barlow packings, two disordered classes and small-period packings. In addition, we compute a lower bound on K for all Barlow packings. We compute Λ¯ for the aforementioned three classes of Barlow packings and find that, to a very good approximation, it is linear in the fraction of fcc-like clusters, taking values between those of least-ordered hcp and most-ordered fcc. This implies that the value of Λ¯ of all Barlow packings is primarily controlled by the local cluster geometry. These results highlight the special nature of anisotropic stacking disorder, which provides impetus for future research on the development of anisotropic order metrics and hyperuniformity properties.

关键词： Classical statistical mechanics

Improving the robustness of the immersed interface method through regularized velocity reconstruction

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arXiv 2025年

作者： Sun, Qi Kolahdouz, Ebrahim M. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

Robust, broadly applicable fluid-structure interaction (FSI) algorithms remain a challenge for computational mechanics. Efforts in this area are driven by the need to enhance predictive accuracy and efficiency in FSI simulations, align with experimental observations, and unravel complex multiscale and multiphysics phenomena, while addressing challenges in developing more robust and efficient methodologies. In previous work, we introduced an immersed interface method (IIM) for discrete surfaces and an extension based on an immersed Lagrangia-Eulerian (ILE) coupling strategy for modeling FSI involving complex geometries. The ability of the method to sharply resolve stress discontinuities induced by singular immersed boundary forces in the presence of low-regularity geometrical representations makes it a compelling choice for three-dimensional modeling of complex geometries in diverse engineering applications. Although the IIM we previously introduced offers many desirable advantages, it also imposes a restrictive mesh factor ratio, requiring the surface mesh to be coarser than the fluid grid to ensure stability. This is because when the mesh factor ratio constraint is not satisfied, parts of the structure motion are not controlled by the discrete FSI system. This constraint can significantly increase computational costs, particularly in applications involving multiscale geometries with highly localized complexity or fine-scale features. To address this limitation, we devise a stabilization strategy for the velocity restriction operator inspired by Tikhonov regularization. This study demonstrates that using a stabilized velocity restriction operator in IIM enables a broader range of structure-to-fluid grid-size ratios without compromising accuracy or altering the flow dynamics. This advancement significantly broadens the applicability of the method to real-world FSI problems involving complex geometries and dynamic conditions, offering a robust and computationally effi

关键词： Mesh generation

An immersed interface method for incompressible flows and geometries with sharp features

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Journal of computational Physics 2025年 537卷

作者： Michael J. Facci Ebrahim M. Kolahdouz Boyce E. Griffith Department of Mathematics University of North Carolina Chapel Hill NC USA Flatiron Institute Simons Foundation New York NY USA Department of Biomedical Engineering University of North Carolina Chapel Hill NC USA Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel Hill NC USA Computational Medicine Program University of North Carolina School of Medicine Chapel Hill NC USA McAllister Heart Institute University of North Carolina School of Medicine Chapel Hill NC USA

The immersed interface method (IIM) for models of fluid flow and fluid-structure interaction imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Most prior work using the IIM for fluid dynamics applications has focused on smooth interfaces, but boundaries with sharp features such as corners and edges can appear in practical analyses, particularly on engineered structures. The present study builds on our work to integrate finite element-type representations of interface geometries with the IIM. Initial realizations of this approach used a continuous Galerkin (CG) finite element discretization for the boundary, but as we show herein, these approaches generate large errors near sharp geometrical features. To overcome this difficulty, this study introduces an IIM approach using a discontinuous Galerkin (DG) representation of the jump conditions. Numerical examples explore the impacts of different interface representations on accuracy for both smooth and sharp boundaries, particularly flows interacting with fixed interface configurations. We demonstrate that using a DG approach provides accuracy that is comparable to the CG method for smooth cases. Further, we identify a time step size restriction for the CG representation that is directly related to the sharpness of the geometry. In contrast, time step size restrictions imposed by DG representations are demonstrated to be nearly insensitive to the presence of sharp features.

关键词： Discrete surface Finite element Fluid-structure interaction Immersed interface method Incompressible flow Jump condition Sharp geometry

Active learning of uniformly accurate interatomic potentials for materials simulation

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Physical Review Materials 2019年第2期3卷 023804-023804页

作者： Linfeng Zhang De-Ye Lin Han Wang Roberto Car Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA and Beijing Institute of Big Data Research Beijing 100871 People's Republic of China

An active learning procedure called deep potential generator (DP-GEN) is proposed for the construction of accurate and transferable machine learning-based models of the potential energy surface (PES) for the molecular modeling of materials. This procedure consists of three main components: exploration, generation of accurate reference data, and training. Application to the sample systems of Al, Mg, and Al-Mg alloys demonstrates that DP-GEN can produce uniformly accurate PES models with a minimal number of reference data.

关键词： Atomic & molecular structure Potential energy surfaces Alloys Amorphous materials Liquids Density functional theory First-principles calculations Molecular dynamics

Assessing the reliability of wind power operations under a changing climate with a non-Gaussian bias correction

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arXiv 2020年

作者： Zhang, Jiachen Crippa, Paola Genton, Marc G. Castruccio, Stefano Department of Applied and Computational Mathematics and Statistics University of Notre Dame United States Department of Civid and Environmental Engineering and Geoscience University of Notre Dame United States Statistics Program King Abdullah University of Science and Technology Saudi Arabia

Facing increasing societal and economic pressure, many countries have established strategies to develop renewable energy portfolios, whose penetration in the market can alleviate the dependence on fossil fuels. In the case of wind, there is a fundamental question related to the resilience, and hence profitability, of future wind farms to a changing climate, given that current wind turbines have lifespans of up to thirty years. In this work we develop a new non-Gaussian method to adjust assimilated observational data to simulations and to estimate future wind, predicated on a trans-Gaussian transformation and a cluster-wise minimization of the Kullback-Leibler divergence. Future winds abundance will be determined for Saudi Arabia, a country with a recently established plan to develop a portfolio of up to 16 GW of wind energy. Further, we estimate the change in profits over future decades using additional high-resolution simulations, an improved method for vertical wind extrapolation and power curves from a collection of popular wind turbines. We find an overall increase in daily profit of $272,000 for the wind energy market for the optimal locations for wind farming in the country. Copyright © 2020, The Authors. All rights reserved.

关键词： Wind power

Generation and structural characterization of Debye random media

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arXiv 2020年

作者： Ma, Zheng Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

In their seminal paper on scattering by an inhomogeneous solid, Debye and coworkers proposed a simple exponentially decaying function for the two-point correlation function of an idealized class of two-phase random media. Such Debye random media, which have been shown to be realizable, are singularly distinct from all other models of two-phase media in that they are entirely defined by their one- and two-point correlation functions. To our knowledge, there has been no determination of other microstructural descriptors of Debye random media. In this paper, we generate Debye random media in two dimensions using an accelerated Yeong-Torquato construction algorithm. We then ascertain microstructural descriptors of the constructed media, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Accurate semi-analytic and empirical formulas for these descriptors are devised. We compare our results for Debye random media to those of other popular models (overlapping disks and equilibrium hard disks), and find that the former model possesses a wider spectrum of hole sizes, including a substantial fraction of large holes. Our algorithm can be applied to generate other models defined by their two-point correlation functions, and their other microstructural descriptors can be determined and analyzed by the procedures laid out here. Copyright © 2020, The Authors. All rights reserved.

关键词： Random processes

Sensitivity kernels for inferring Lorentz stresses from normal-mode frequency splittings in the Sun

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arXiv 2020年

作者： Das, Srijan Bharati Chakraborty, Tuneer Hanasoge, Shravan M. Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ United States Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Geosciences and Program in Applied & Computational Mathematics Princeton University Princeton NJ United States

Departures from standard spherically symmetric solar models, in the form of perturbations such as global and local-scale flows and structural asphericities, result in the splitting of eigenfrequencies in the observed spectrum of solar oscillations. Drawing from prevalent ideas in normal-mode coupling theory in geophysical literature, we devise a procedure that enables the computation of sensitivity kernels for general Lorentz stress fields in the Sun. Mode coupling due to any perturbation requires careful consideration of self- and cross-coupling of multiplets. Invoking the isolated-multiplet approximation allows for limiting the treatment to purely self-coupling, requiring significantly less computational resources. We identify the presence of such isolated multiplets under the effect of Lorentz stresses in the Sun. Currently, solar missions allow precise measurements of self-coupling of multiplets via "a-coefficients" and the cross-spectral correlation signal which enables the estimation of the "structure coefficients". We demonstrate the forward problem for both self-coupling (a-coefficients) and cross-coupling (structure coefficients). In doing so, we plot the self-coupling kernels and estimate a-coefficients arising from a combination of deep-toroidal and surface-dipolar axisymmetric fields. We also compute the structure coefficients for an arbitrary general magnetic field (real and solenoidal) and plot the corresponding "splitting function", a convenient way to visualize the splitting of multiplets under 3D internal perturbations. The results discussed in this paper pave the way to formally pose an inverse problem, and infer solar internal magnetic fields. Copyright © 2020, The Authors. All rights reserved.

关键词： Chemical reactions