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Inverting the markovian projection, with an application to local stochastic volatility models

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arXiv 2019年

作者： Lacker, Daniel Shkolnikov, Mykhaylo Zhang, Jiacheng Ieor Department Columbia University New YorkNY10027 United States Orfe Department Bendheim Center for Finance Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 Orfe Department Princeton University PrincetonNJ08544 United States

We study two-dimensional stochastic differential equations (SDEs) of McKean- Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed in [Gyö86], typically to produce an It o process with the fixed-time marginal distributions of a given one-dimensional diffusion but richer dynamical features. We prove the strong existence of stationary solutions for these SDEs, as well as their strong uniqueness in an important special case. Variants of the SDEs discussed in this paper enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding. Copyright © 2019, The Authors. All rights reserved.

关键词： Stochastic systems

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

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Physical Review E 2018年第1期97卷 012118-012118页

作者： Michael A. Klatt Salvatore 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 Princeton New Jersey 08544 USA

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

关键词： Classical statistical mechanics

Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications

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Physical Review E 2018年第1期98卷 013307-013307页

作者： Zheng Ma Salvatore 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 Princeton New Jersey 08544 USA

The quantitative characterization of the microstructure of random heterogeneous media in d-dimensional Euclidean space Rd via a variety of n-point correlation functions is of great importance, since the respective infinite set determines the effective physical properties of the media. In particular, surface-surface Fss and surface-void Fsv correlation functions (obtainable from radiation scattering experiments) contain crucial interfacial information that enables one to estimate transport properties of the media (e.g., the mean survival time and fluid permeability) and complements the information content of the conventional two-point correlation function. However, the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use. We first present a concise derivation of the small-r behaviors of these functions, which are linked to the mean curvature of the system. Then we demonstrate that one can reduce the computational complexity of the problem, without sacrificing accuracy, by extracting the necessary interfacial information from a cut of the d-dimensional statistically homogeneous and isotropic system with an infinitely long line. Accordingly, we devise algorithms based on this idea and test them for two-phase media in continuous and discrete spaces. Specifically for the exact benchmark model of overlapping spheres, we find excellent agreement between numerical and exact results. We compute surface correlation functions and corresponding local surface-area variances for a variety of other model microstructures, including hard spheres in equilibrium, decorated “stealthy” patterns, as well as snapshots of evolving pattern formation processes (e.g., spinodal decomposition). It is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures.

关键词： Structural properties

Multiscale PHATE Exploration of SARS-CoV-2 Data Reveals Multimodal Signatures of Disease

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Research Square

Research Square 2021年

作者： Kuchroo, Manik Huang, Jessie Wong, Patrick Grenier, Jean-Christophe Shung, Dennis Tong, Alexander Lucas, Carolina Klein, Jon Burkhardt, Daniel B. Gigante, Scott Godavarthi, Abhinav Rieck, Bastian Israelow, Benjamin Simonov, Michael Mao, Tianyang Oh, Ji Eun Silva, Julio Takahashi, Takehiro Odio, Camila D. Casanovas-Massana, Arnau Fournier, John Farhadian, Shelli Dela Cruz, Charles S. Ko, Albert I. Hirn, Matthew J. Wilson, F. Perry Hussin, Julie Wolf, Guy Iwasaki, Akiko Krishnaswamy, Smita Department of Neuroscience Yale University New HavenCT United States Department of Computer Science Yale University New HavenCT United States Department of Immunobiology Yale University New HavenCT United States Montreal Heart Institute MontréalQC Canada Department of Medicine Yale University New HavenCT United States Department of Genetics Yale University New HavenCT United States Computational Biology Bioinformatics Program Yale University New HavenCT United States Department of Applied Mathematics Yale University New HavenCT United States Department of Biosystems Science and Engineering ETH Zurich Switzerland Department of Epidemiology of Microbial Diseases Yale School of Public Health New HavenCT United States Department of Medicine Section of Infectious Diseases Yale University School of Medicine New HavenCT United States Department of Medicine Section of Pulmonary and Critical Care Medicine Yale University School of Medicine New HavenCT United States Department of Computational Mathematics Science and Engineering Michigan State University East LansingMI United States Department of Mathematics Michigan State University East LansingMI United States Clinical and Translational Research Accelerator Department of Medicine Yale University New HavenCT United States Faculty of Medicine Université de Montréal Québec Canada Mila – Quebec AI institute MontréalQC Canada Department of Mathematics and Statistics Université de Montréal MontréalQC Canada Howard Hughes Medical Institute Chevy ChaseMD United States

The biomedical community is producing increasingly high dimensional datasets, integrated from hundreds of patient samples, which current computational techniques struggle to explore. To uncover biological meaning from these complex datasets, we present an approach called Multiscale PHATE, which learns abstracted biological features from data that can be directly predictive of disease. Built on a coarse graining process called diffusion condensation, Multiscale PHATE learns a data topology that can be analyzed at coarse levels for high level summarizations of data, as well as at fine levels for detailed representations on subsets. We apply Multiscale PHATE to study the immune response to COVID-19 in 54 million cells from 168 hospitalized patients. Through our analysis of patient samples, we identify CD16hiCD66blo neutrophil and IFNγ+GranzymeB+ Th17 cell responses enriched in patients who die. Furthermore, we show that population groupings Multiscale PHATE discovers can be directly fed into a classifier to predict disease outcome. We also use Multiscale PHATE-derived features to construct two different manifolds of patients, one from abstracted flow cytometry features and another directly on patient clinical features, both associating immune subsets and clinical markers with outcome. © 2021, CC BY.

关键词： Coronavirus

Active learning of uniformly accurate inter-atomic potentials for materials simulation

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arXiv 2018年

作者： Zhang, Linfeng Lin, De-Ye Wang, Han Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China CAEP Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 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. Copyright © 2018, The Authors. All rights reserved.

关键词： Potential energy

Deterministic guarantees for Burer–Monteiro factorizations of smooth semidefinite programs

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arXiv 2018年

作者： Boumal, Nicolas Voroninski, Vladislav Bandeira, Afonso S. Mathematics Department Program in Applied and Computational Mathematics Princeton University Department of Mathematics Center for Data Science Courant Institute of Mathematical Sciences New York University

We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer–Monteiro approach, we optimize a factor Y of size n× p instead, such that X = YY. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a non-convex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite non-convexity, first- and second-order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YY of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust-region subproblem and quadratic optimization over several spheres, as well as for the Max-Cut and Orthogonal-Cut SDPs which are common relaxations in stochastic block modeling and synchronization of rotations. Copyright © 2018, The Authors. All rights reserved.

关键词： Stochastic systems

Adaptive coupling of a deep neural network potential to a classical force field

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arXiv 2018年

作者： Zhang, Linfeng Wang, Han Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Data Science and Beijing International Center for Mathematical Research Peking University China Beijing Institute of Big Data Research Beijing100871 China

An adaptive modeling method (AMM) that couples a deep neural network potential and a classical force field is introduced to address the accuracy-efficiency dilemma faced by the molecular simulation community. The AMM simulated system is decomposed into three types of regions. The first type captures the important phenomena in the system and requires high accuracy, for which we use the Deep Potential Molecular Dynamics (DeePMD) model in this work. The DeePMD model is trained to accurately reproduce the statistical properties of the ab initio molecular dynamics. The second type does not require high accuracy and a classical force field is used to describe it in an efficient way. The third type is used for a smooth transition between the first and the second types of regions. By using a force interpolation scheme and imposing a thermodynamics force in the transition region, we make the DeePMD region embedded in the AMM simulated system as if it were embedded in a system that is fully described by the accurate potential. A representative example of the liquid water system is used to show the feasibility and promise of this method. Copyright © 2018, The Authors. All rights reserved.

关键词： Deep neural networks

DeePCG: Constructing coarse-grained models via deep neural networks

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arXiv 2018年

作者： Weinan, E. Zhang, Linfeng Han, Jiequn Wang, Han Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China Caep Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Program in Applied and Computational Mathematics Princeton Institute for Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

We introduce a general framework for constructing coarse-grained potential models without ad hoc approximations such as limiting the potential to two- A nd/or three-body contributions. The scheme, called Deep Coarse-Grained Potential (abbreviated DeePCG), exploits a carefully crafted neural network to construct a many-body coarse-grained potential. The network is trained with full atomistic data in a way that preserves the natural symmetries of the system. The resulting model is very accurate and can be used to sample the configurations of the coarse-grained variables in a much faster way than with the original atomistic model. As an application we consider liquid water and use the oxygen coordinates as the coarse-grained variables, starting from a full atomistic simulation of this system at the ab initio molecular dynamics level. We find that the two-body, three-body, and higher-order oxygen correlation functions produced by the coarse-grained and full atomistic models agree very well with each other, illustrating the effectiveness of the DeePCG model on a rather challenging task. Copyright © 2018, The Authors. All rights reserved.

关键词： Deep neural networks

How SGD selects the global minima in over-parameterized learning: a dynamical stability perspective 18

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How SGD selects the global minima in over-parameterized lear...

Proceedings of the 32nd International Conference on Neural Information Processing Systems

作者： Lei Wu Chao Ma Weinan E. School of Mathematical Sciences Peking University Beijing P.R. China Program in Applied and Computational Mathematics Princeton University Princeton NJ Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton NJ and Beijing Institute of Big Data Research Beijing P.R. China

The question of which global minima are accessible by a stochastic gradient decent (SGD) algorithm with specific learning rate and batch size is studied from the perspective of dynamical stability. The concept of non-uniformity is introduced, which, together with sharpness, characterizes the stability property of a global minimum and hence the accessibility of a particular SGD algorithm to that global minimum. In particular, this analysis shows that learning rate and batch size play different roles in minima selection. Extensive empirical results seem to correlate well with the theoretical findings and provide further support to these claims.

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