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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

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

Physics-informed machine learning with smoothed particle hydrodynamics: Hierarchy of reduced Lagrangian models of turbulence

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Physical Review Fluids 2023年第5期8卷 054602-054602页

作者： Michael Woodward Yifeng Tian Criston Hyett Chris Fryer Mikhail Stepanov Daniel Livescu Michael Chertkov Graduate Interdisciplinary Program in Applied Mathematics University of Arizona Tucson Arizona 85721 USA Department of Mathematics University of Arizona Tucson Arizona 85721 USA Computer Computational and Statistical Sciences Division LANL Los Alamos New Mexico 87544 USA

Building efficient, accurate, and generalizable reduced-order models of developed turbulence remains a major challenge. This manuscript approaches this problem by developing a hierarchy of parameterized reduced Lagrangian models for turbulent flows, and it investigates the effects of enforcing physical structure through smoothed particle hydrodynamics (SPH) versus relying on neural networks (NNs) as universal function approximators. Starting from NN parametrizations of a Lagrangian acceleration operator, this hierarchy of models gradually incorporates a weakly compressible and parameterized SPH framework, which enforces physical symmetries, such as Galilean, rotational, and translational invariances. Within this hierarchy, two new parameterized smoothing kernels are developed to increase the flexibility of the learn-able SPH simulators. For each model we experiment with different loss functions which are minimized using gradient based optimization, where efficient computations of gradients are obtained by using automatic differentiation and sensitivity analysis. Each model within the hierarchy is trained on two data sets associated with weakly compressible homogeneous isotropic turbulence: (1) a validation set using weakly compressible SPH; and (2) a high-fidelity set from direct numerical simulations. Numerical evidence shows that encoding more SPH structure improves generalizability to different turbulent Mach numbers and time shifts, and that including the novel parameterized smoothing kernels improves the accuracy of SPH at the resolved scales.

关键词： Sensitivity analysis

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

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Physical Review B 2021年第2期103卷 024304-024304页

作者： Pinchen Xie Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University New Jersey 08544 USA

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 that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits 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 one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

关键词： Quantum quench Quantum statistical mechanics Nonequilibrium lattice models Machine learning Monte Carlo methods

Learning algorithms for mean field optimal control

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arXiv 2025年

作者： Soner, H. Mete Teichmann, Josef Yan, Qinxin Department of Operations Research and Financial Engineering Princeton University PrincetonNJ08540 United States Department of Mathematics ETH Zürich Switzerland Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

We analyze an algorithm to numerically solve the mean-field optimal control problems by approximating the optimal feedback controls using neural networks with problem specific architectures. We approximate the model by an N-particle system and leverage the exchangeability of the particles to obtain substantial computational efficiency. In addition to several numerical examples, a convergence analysis is provided. We also developed a universal approximation theorem on Wasserstein *** Codes 35Q89, 35D40, 49L25, 60G99 © 2025, CC BY-NC-ND.

关键词： Optimal control systems

A Stochastic Approximate Expectation-Maximization for Structure Determination Directly from Cryo-EM Micrographs

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arXiv 2023年

作者： Kreymer, Shay Singer, Amit Bendory, Tamir The School of Electrical Engineering Tel Aviv University Tel Aviv Israel The Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

A single-particle cryo-electron microscopy (cryoEM) measurement, called a micrograph, consists of multiple two-dimensional tomographic projections of a three-dimensional molecular structure at unknown locations, taken under unknown viewing directions. All existing cryo-EM algorithmic pipelines first locate and extract the projection images, and then reconstruct the structure from the extracted images. However, if the molecular structure is small, the signal-to-noise ratio (SNR) of the data is very low, and thus accurate detection of projection images within the micrograph is challenging. Consequently, all standard techniques fail in low-SNR regimes. To recover molecular structures from measurements of low SNR, and in particular small molecular structures, we devise a stochastic approximate expectation-maximization algorithm to estimate the three-dimensional structure directly from the micrograph, bypassing locating the projection images. We corroborate our computational scheme with numerical experiments, and present successful structure recoveries from simulated noisy measurements. © 2023, CC BY.

关键词： Molecular structure

Neural network representation of electronic structure from ab initio molecular dynamics

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Science Bulletin 2022年第1期67卷 29-37,M0003页

作者： Qiangqiang Gu Linfeng Zhang Ji Feng International Center for Quantum Materials School of PhysicsPeking UniversityBeijing 100871China Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Collaborative Innovation Center of Quantum Matter Beijing 100871China

Despite their rich information content,electronic structure data amassed at high volumes in ab initio molecular dynamics simulations are generally *** introduce a transferable high-fidelity neural network representation of such data in the form of tight-binding Hamiltonians for crystalline *** predictive representation of ab initio electronic structure,combined with machinelearning boosted molecular dynamics,enables efficient and accurate electronic evolution and *** it is applied to a one-dimension charge-density wave material,carbyne,we are able to compute the spectral function and optical conductivity in the canonical *** spectral functions evaluated during soliton-antisoliton pair annihilation process reveal significant renormalization of low-energy edge modes due to retarded electron-lattice coupling beyond the Born-Oppenheimer *** availability of an efficient and reusable surrogate model for the electronic structure dynamical system will enable calculating many interesting physical properties,paving the way to previously inaccessible or challenging avenues in materials modeling.

关键词： Neural network Tight-binding model Electronic structure ab initio molecular dynamics

Can Shallow Neural Networks Beat the Curse of Dimensionality? A Mean Field Training Perspective

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IEEE Transactions on Artificial Intelligence

IEEE Transactions on Artificial Intelligence 2020年第2期1卷 121-129页

作者： Wojtowytsch, Stephan Weinan, E. The Program of Applied and Computational Mathematics Princeton University Princeton 08544 NJ United States The Department of Mathematics The Program of Applied and Computational Mathematics Princeton University Princeton 08544 NJ United States

We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than t−4/(d−2) under mean field scaling. The loss functiona... 详细信息

We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than t⁻⁴/(^d−2) under mean field scaling. The loss functional is mean squared error with a Lipschitz-continuous target function and data distributed uniformly on the d-dimensional unit cube. Thus gradient descent training for fitting reasonably smooth, but truly high-dimensional data may be subject to the curse of dimensionality. We present numerical evidence that gradient descent training with general Lipschitz target functions becomes slower and slower as the dimension increases, but converges at approximately the same rate in all dimensions when the target function lies in the natural function space for two-layer ReLU networks. Impact Statement–Artificial neural networks perform well in many real life applications, but may suffer from the curse of dimensionality on certain problems. We provide theoretical and numerical evidence that this may be related to whether a target function lies in the hypothesis class described by infinitely wide networks. The training dynamics are considered in the fully non-linear regime and not reduced to neural tangent kernels. We believe that it will be essential to study these hypothesis classes in detail to choose an appropriate machine learning models for a given problem. The goal of the article is to illustrate this in a mathematically sound and numerically convincing fashion. © 2021 IEEE.

关键词： Artificial neural networks classification and regression supervised learning

STEFAN PROBLEM WITH SURFACE TENSION: UNIQUENESS OF PHYSICAL SOLUTIONS UNDER RADIAL SYMMETRY

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arXiv 2023年

作者： Guo, Yucheng Nadtochiy, Sergey Shkolnikov, Mykhaylo Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We study the Stefan problem with surface tension and radially symmetric initial data. In this context, the notion of a so-called physical solution, which exists globally despite the inherent blow-ups of the melting rate, has been recently introduced in [NS23]. The paper at hand is devoted to the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. In the course of the proof, we establish a wide variety of regularity estimates for the free boundary and for the temperature function, of interest in their own *** Codes 80A22, 35B44, 60H30, 35B05 Copyright © 2023, The Authors. All rights reserved.

关键词： Surface tension