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On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers

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arXiv 2020年

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

A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if h(x) = Af(x) + b where A is a linear map and f is the output of the penultimate layer of the network (after activation), then all data points xi1,..., xi,Ni in a class Ci are mapped to a single point yi by f and the points yi are located at the vertices of a regular k − 1-dimensional standard simplex in a high-dimensional Euclidean space. We explain this observation analytically in toy models for highly expressive deep neural networks. In complementary examples, we demonstrate rigorously that even the final output of the classifier h is not uniform over data samples from a class Ci if h is a shallow network (or if the deeper layers do not bring the data samples into a convenient geometric configuration).MSC Codes 68T07, 62H30 Copyright © 2020, The Authors. All rights reserved.

关键词： Deep neural networks

THE COMPLEX INTERPLAY BETWEEN RISK TOLERANCE AND THE SPREAD OF INFECTIOUS DISEASES

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arXiv 2024年

作者： Nguyen, Maximilian Freedman, Ari Cheung, Matthew Saad-Roy, Chadi Espinoza, Baltazar Grenfell, Bryan Levin, Simon Lewis-Sigler Institute Princeton University PrincetonNJ08544 United States Department of Ecology and Evolutionary Biology Princeton University PrincetonNJ United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Miller Institute for Basic Research in Science Department of Integrative Biology University of California Berkeley BerkeleyCA United States Biocomplexity Institute University of Virginia CharlottesvilleVA United States

Risk-driven behavior provides a feedback mechanism through which individuals both shape and are collectively affected by an epidemic. We introduce a general and flexible compartmental model to study the effect of heterogeneity in the population with regards to risk tolerance. The interplay between behavior and epidemiology leads to a rich set of possible epidemic dynamics. Depending on the behavioral composition of the population, we find that increasing heterogeneity in risk tolerance can either increase or decrease the epidemic size. We find that multiple waves of infection can arise due to the interplay between transmission and behavior, even without the replenishment of susceptibles. We find that increasing protective mechanisms such as the effectiveness of interventions, the number of risk-averse people in the population, and the duration of intervention usage reduces the epidemic overshoot. When the protection is pushed past a critical threshold, the epidemic dynamics enter an underdamped regime where the epidemic size exactly equals the herd immunity threshold. Lastly, we can find regimes where epidemic size does not monotonically decrease with a population that becomes increasingly risk-averse. Copyright © 2024, The Authors. All rights reserved.

关键词： Epidemiology

Some observations on high-dimensional partial differential equations with barron data

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arXiv 2020年

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

We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under *** Codes 68T07, 35C15, 65M80 Copyright © 2020, The Authors. All rights reserved.

关键词： Neural networks

A priori estimates for classification problems using neural networks

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arXiv 2020年

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

We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and direct approximation theorems to obtain a priori error estimates for regularized loss *** Codes 68T07, 60-08 Copyright © 2020, The Authors. All rights reserved.

关键词： Classification (of information)

Three-Dimensional Construction of Hyperuniform, Nonhyperuniform and Antihyperuniform Random Media via Spectral Density Functions and Their Transport Properties

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arXiv 2024年

作者： Shi, Wenlong Jiao, Yang Torquato, Salvatore Arizona State University TempeAZ85287 United States Department of Physics Arizona State University TempeAZ85287 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function χ˜V(k), which can be obtained experimentally from scattering techniques, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability S(t) and effective dynamic dielectric constant ϵe for electromagnetic wave propagation. Moreover, χ˜V(k) has been employed to derive sharp bounds on the fluid permeability k. Given the importance of χ˜V(k), we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical χ˜V(k) functions that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions . We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by a power-law autocovariance function χV (r) and contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability S(t) and estimate the fluid permeability k associated with all of the constructed materials directly from the corresponding χ˜V(k) functions. Although it is well established that the long-time asymptotic scaling behavior of S(t) only depends on the functional form of χ˜V(k), with the stealthy hyperuniform and antihyperuniform media respectively

关键词： Spectral density

On the banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics

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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 United States

We develop Banach spaces for ReLU neural networks of finite depth L and infinite width. The spaces contain all finite fully connected L-layer networks and their L2-limiting objects under bounds on the natural path-norm. Under this norm, the unit ball in the space for L-layer networks has low Rademacher complexity and thus favorable generalization properties. Functions in these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates. The key to this work is a new way of representing functions in some form of expectations, motivated by multi-layer neural networks. This representation allows us to define a new class of continuous models for machine learning. We show that the gradient flow defined this way is the natural continuous analog of the gradient descent dynamics for the associated multi-layer neural networks. We show that the path-norm increases at most polynomially under this continuous gradient flow *** Codes 68T07, 46E15, 26B35, 26B40 Copyright © 2020, The Authors. All rights reserved.

关键词： Deep neural networks

Efficient sampling of high-dimensional free energy landscapes using adaptive reinforced dynamics

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arXiv 2021年

作者： Wang, Dongdong Zhang, Linfeng Wang, Yanze Chang, Junhan Wang, Han Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States DP Technology Beijing China College of Chemistry and Molecular Engineering Peking University Beijing100871 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China School of Mathematical Sciences Peking University Beijing China Department of Mathematics Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

Enhanced sampling methods such as metadynamics and umbrella sampling have become essential tools for exploring the configuration space of molecules and materials. At the same time, they have long faced a number of issues such as the inefficiency when dealing with a large number of collective variables (CVs) or systems with high free energy barriers. In this work, we show that with the clustering and adaptive tuning techniques, the reinforced dynamics (RiD) scheme can be used to efficiently explore the configuration space and free energy landscapes with a large number of CVs or systems with high free energy barriers. We illustrate this by studying various representative and challenging examples. Firstly we demonstrate the efficiency of adaptive RiD compared with other methods, and construct the 9-dimensional free energy landscape of peptoid trimer which has energy barriers of more than 8 kcal/mol. We then study the folding of the protein chignolin using 18 CVs. In this case, both the folding and unfolding rates are observed to be equal to 4.30 µs−1. Finally, we propose a protein structure refinement protocol based on RiD. This protocol allows us to efficiently employ more than 100 CVs for exploring the landscape of protein structures and it gives rise to an overall improvement of 14.6 units over the initial Global Distance Test-High Accuracy (GDT-HA) score. © 2021, CC BY.

关键词： Wave functions

An immersed peridynamics model of fluid-structure interaction accounting for material damage and failure

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arXiv 2022年

作者： Kim, Keon Ho Bhalla, Amneet P.S. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Mechanical Engineering San Diego State University San DiegoCA United States Departments of Mathematics Applied Physical Sciences and 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 Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States

This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary methods. The major difference between our approach and conventional immersed boundary methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well-characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations. © 2022, CC BY-SA.

关键词： Continuum mechanics

An Immersed Peridynamics Model of Fluid-Structure Interaction Accounting for Material Damage and Failure

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SSRN

SSRN 2022年

作者： Kim, Keon Ho Bhalla, Amneet Pal Singh Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Mechanical Engineering San Diego State University San DiegoCA United States Departments of Mathematics Applied Physical Sciences and 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 Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States

This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary (IB) methods. The major difference between our approach and conventional IB methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations. © 2022, The Authors. All rights reserved.

关键词： Fluid structure interaction