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arXiv 2019年

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

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are uniformly close to that of a kernel method. For general values of the network width and training data size, sharp estimates of the generalization error is established for target functions in the appropriate reproducing kernel Hilbert *** Codes 41A99, 49M99 Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods

Cosserat elasticity as the weak-field limit of Einstein-Cartan relativity

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Physical Review D 2024年第10期109卷 104052-104052页

作者： Matthew Maitra Jeroen Tromp Institut für Geophysik ETH Zürich Sonneggstrasse 5 Zürich 8092 Switzerland Department of Geosciences Program in Applied and Computational Mathematics Princeton University Princeton 08540 NJ United States

The weak-field limit of Einstein-Cartan (EC) relativity is studied. The equations of EC theory are rewritten such that they formally resemble those of Einstein general relativity (EGR); this allows ideas from post-Newtonian theory to be imported without essential change. The equations of motion are then written both at first post-Newtonian (1PN) order and at 1.5PN order. EC theory’s 1PN equations of motion are found to be those of a micropolar/Cosserat elastic medium, along with a decoupled evolution equation for nonclassical, spin-related fields. It seems that a necessary condition for these results to hold is that one chooses the nonclassical fields to scale with the speed of light in a certain empirically reasonable way. Finally, the 1.5PN equations give greater insight into the coupling between energy-momentum and spin within slowly moving, weakly gravitating matter. Specifically, the weakly relativistic modifications to Cosserat theory involve a gravitational torque and an augmentation of the gravitational force due to a dynamic mass moment density with an accompanying dynamic mass moment density flux, and new forms of linear momentum density captured by a dynamic mass density flux and a dynamic momentum density.

关键词： Alternative gravity theories Solar system & its planets

Fast wavelet-based single-particle reconstruction in Cryo-EM

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Fast wavelet-based single-particle reconstruction in Cryo-EM

IEEE International Symposium on Biomedical Imaging

作者： Cédric Vonesch Lanhui Wang Yoel Shkolnisky Amit Singer Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Department of Applied Mathematics Tel-Aviv University Tel-Aviv Israel

This paper presents a novel algorithm for the 3D tomographic inversion problem that arises in single-particle electron cryomicroscopy (Cryo-EM). It is based on two key components: 1) a variational formulation that promotes sparsity in the wavelet domain and 2) the Toeplitz structure of the combined projection/back-projection operator. The first idea has proven to be very effective for the recovery of piecewise-smooth signals, which is confirmed by our numerical experiments. The second idea allows for a computationally efficient implementation of the reconstruction procedure, using only one circulant convolution per iteration.

关键词： Image reconstruction Signal to noise ratio Three dimensional displays Fourier transforms Kernel Tomography

A Priori Estimates of the Population Risk for Residual Networks

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arXiv 2019年

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

Optimal a priori estimates are derived for the population risk, also known as the generalization error, of a regularized residual network model. An important part of the regularized model is the usage of a new path norm, called the weighted path norm, as the regularization term. The weighted path norm treats the skip connections and the nonlinearities differently so that paths with more nonlinearities are regularized by larger weights. The error estimates are a priori in the sense that the estimates depend only on the target function, not on the parameters obtained in the training process. The estimates are optimal, in a high dimensional setting, in the sense that both the bound for the approximation and estimation errors are comparable to the Monte Carlo error rates. A crucial step in the proof is to establish an optimal bound for the Rademacher complexity of the residual networks. Comparisons are made with existing norm-based generalization error bounds. Copyright © 2019, The Authors. All rights reserved.

关键词： Error analysis

Machine learning from a continuous viewpoint

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arXiv 2019年

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

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, very much in the spirit of classical numerical analysis and statistical physics. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the shallow neural network model and the residual neural network model, can all be recovered as particular discretizations of different continuous formulations. We also present examples of new models, such as the ow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework. Copyright © 2019, The Authors. All rights reserved.

关键词： Machine learning

ZERO KINETIC UNDERCOOLING LIMIT IN THE SUPERCOOLED STEFAN PROBLEM

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arXiv 2020年

作者： Baker, Graeme Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We study the solutions of the one-phase supercooled Stefan problem with kinetic undercooling, which describes the freezing of a supercooled liquid, in one spatial dimension. Assuming that the initial temperature lies between the equilibrium freezing point and the characteristic invariant temperature throughout the liquid our main theorem shows that, as the kinetic undercooling parameter tends to zero, the free boundary converges to the (possibly irregular) free boundary in the supercooled Stefan problem without kinetic undercooling, whose uniqueness has been recently established in [DNS19], [LS18b]. The key tools in the proof are a Feynman-Kac formula, which expresses the free boundary in the problem with kinetic undercooling through a local time of a reflected process, and a resulting comparison principle for the free boundaries with different kinetic undercooling *** Codes 80A22, 35B44, 60H30 Copyright © 2020, The Authors. All rights reserved.

关键词： Supercooling

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

Isotope effects in x-ray absorption spectra of liquid water

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Physical Review B 2020年第11期102卷 115155-115155页

作者： Chunyi Zhang Linfeng Zhang Jianhang Xu Fujie Tang Biswajit Santra Xifan Wu Department of Physics Temple University Philadelphia Pennsylvania 19122 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The isotope effects in x-ray absorption spectra of liquid water are studied by a many-body approach within electron-hole excitation theory. The molecular structures of both light and heavy water are modeled by path-integral molecular dynamics based on the advanced deep-learning technique. The neural network is trained on ab initio data obtained with SCAN density functional theory. The experimentally observed isotope effect in x-ray absorption spectra is reproduced semiquantitatively in theory. Compared to the spectrum in normal water, the blueshifted and less pronounced pre- and main-edge in heavy water reflect that the heavy water is more structured at short- and intermediate-range of the hydrogen-bond network. In contrast, the isotope effect on the spectrum is negligible at post-edge, which is consistent with the identical long-range ordering in both liquids as observed in the diffraction experiment.

关键词： Atomic & molecular structure Electronic excitation & ionization Water Density functional calculations Electronic structure First-principles calculations Machine learning Molecular dynamics X-ray absorption spectroscopy

A SINGULAR TWO-PHASE STEFAN PROBLEM AND PARTICLES INTERACTING THROUGH THEIR HITTING TIMES

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arXiv 2022年

作者： Baker, Graeme Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance and Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension that can be found in [DIRT15a], [NS19a], [HLS18], [CRS20]. As therein, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not monotone, so that the large system limit is obtained by a novel *** Codes 82C22, 60H30, 35K10, 35K60, 91G80 Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

WELL-POSEDNESS OF THE SUPERCOOLED STEFAN PROBLEM WITH OSCILLATORY INITIAL CONDITIONS

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arXiv 2023年

作者： Mustapha, Scander Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance and Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We study the one-phase one-dimensional supercooled Stefan problem with oscillatory initial conditions. In this context, the global existence of so-called physical solutions has been shown recently in [CRSF23], despite the presence of blow-ups in the freezing rate. On the other hand, for regular initial conditions, the uniqueness of physical solutions has been established in [DNS22]. Here, we prove the uniqueness of physical solutions for oscillatory initial conditions by a new contraction argument that replaces the local monotonicity condition of [DNS22] by an averaging condition. We verify this weaker condition for fairly general oscillating probability densities, such as the ones given by an almost sure trajectory of (1 + Wx - √2x| log | log x||)+ ^ 1 near the origin, where W is a standard Brownian motion. We also permit typical deterministically constructed oscillating densities, including those of the form (1 + sin 1/x)/2 near the origin. Finally, we provide an example of oscillating densities for which it is possible to go beyond our main assumption via further complementary *** Codes 80A22, 35B05, 60H30 Copyright © 2023, The Authors. All rights reserved.

关键词： Brownian movement