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Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence

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arXiv 2022年

作者： Han, Jiequn Hu, Ruimeng Long, Jihao Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States Department of Mathematics Department of Statistics and Applied Probability University of California Santa BarbaraCA93106-3080 United States The Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States

One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus is inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs’ model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration’s FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics (IPMs) previously developed in [Han, Hu and Long, arXiv:2104.12036]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker-Smale model whose cost depends on the full distribution of the forward *** Codes 65C30, 68T07, 49N80, 68Q25 © 2022, CC BY.

关键词： Stochastic systems

CEMRACS 2017: numerical probabilistic approach to MFG

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arXiv 2018年

作者： Angiuli, Andrea Graves, Christy V. Li, Houzhi Chassagneux, Jean-François Delarue, François Carmona, René Department of Statistics and Applied Probability University of California Santa Barbara Program in Applied and Computational Mathematics Princeton University Laboratoire de Probabilités Modélisation et Statistiques Université Paris Diderot Laboratoire Jean-Alexandre Dieudonné Université de Nice Sophia-Antipolis Operations Research and Financial Engineering Bendheim Center for Finance Princeton University

This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [27], and by Huang, Caines, and Malhamé, see [24]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme;importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges. Copyright © 2018, The Authors. All rights reserved.

关键词： Numerical methods

A distribution-free smoothed combination method of biomarkers to improve diagnostic accuracy in multi-category classification

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arXiv 2019年

作者： Maiti, Raju Li, Jialiang Das, Priyam Feng, Lei Hausenloy, Derek Chakraborty, Bibhas Centre for Quantitative Medicine Duke-National University of Singapore Medical School Singapore Department of Statistics and Applied Probability National University of Singapore Singapore Singapore Eye Research Institute Singapore Department of Biostatistics University of Texas MD Anderson Cancer Center United States Department of Psychological Medicine Yong Loo Lin School of Medicine National University of Singapore Singapore Cardiovascular and Metabolic Disorders Program Duke-National University of Singapore Medical School Singapore Department of Biostatistics and Bioinformatics Duke University United States

Results from multiple diagnostic tests are usually combined to improve the overall diagnostic accuracy. For binary classification, maximization of the empirical estimate of the area under the receiver operating characteristic (ROC) curve is widely adopted to produce the optimal linear combination of multiple biomarkers. In the presence of large number of biomarkers, this method proves to be computationally expensive and difficult to implement since it involves maximization of a discontinuous, non-smooth function for which gradient-based methods cannot be used directly. Complexity of this problem increases when the classification problem becomes multi-category. In this article, we develop a linear combination method that maximizes a smooth approximation of the empirical Hypervolume Under Manifolds (HUM) for multi-category outcome. We approximate HUM by replacing the indicator function with the sigmoid function or normal cumulative distribution function (CDF). With the above smooth approximations, efficient gradient-based algorithms can be employed to obtain better solution with less computing time. We show that under some regularity conditions, the proposed method yields consistent estimates of the coefficient parameters. We also derive the asymptotic normality of the coefficient estimates. We conduct extensive simulations to examine our methods. Under different simulation scenarios, the proposed methods are compared with other existing methods and are shown to outperform them in terms of diagnostic accuracy. The proposed method is illustrated using two real medical data sets. Copyright © 2019, The Authors. All rights reserved.

关键词： Biomarkers

Erratum To: Respiratory viral pathogens among Singapore military servicemen 2009 – 2012: epidemiology and clinical characteristics

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BMC Infectious Diseases 2014年第1期14卷 1-2页

作者： Xin Quan Tan Vernon J Lee Xiahong Zhao Mark I-Cheng Chen Alex Richard Cook Jin Phang Loh Boon Huan Tan Wee Hong Victor Koh Sock Hoon Ng Biodefence Centre Ministry of Defence Singapore Singapore Preventive Medicine Residency Programme National University Health System Singapore Singapore Saw Swee Hock School of Public Health National University of Singapore Singapore Singapore Department of Clinical Epidemiology Tan Tock Seng Hospital Singapore Singapore Yale-NUS College National University of Singapore Singapore Singapore Program in Health Services and Systems Research Duke-NUS Graduate Medical School Singapore Singapore Department of Statistics and Applied Probability National University of Singapore Singapore Singapore Defence Medical and Environmental Research Institute Singapore Singapore

DeePN2: A deep learning-based non-Newtonian hydrodynamic model

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arXiv 2021年

作者： Fang, Lidong Ge, Pei Zhang, Lei Weinan, E. Lei, Huan Department of Computational Mathematics Science and Engineering Michigan State University MI48824 United States School of Mathematical Sciences Institute of Natural Sciences and MOE-LSC Shanghai Jiao Tong University 800 Dongchuan Road Shanghai200240 China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing100871 China AI for Science Institute Beijing100080 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States Department of Statistics and Probability Michigan State University MI48824 United States

A long standing problem in the modeling of non-Newtonian hydrodynamics of polymeric flows is the availability of reliable and interpretable hydrodynamic models that faithfully encode the underlying micro-scale polymer dynamics. The main complication arises from the long polymer relaxation time, the complex molecular structure and heterogeneous interaction. DeePN2, a deep learning-based non-Newtonian hydrodynamic model, has been proposed and has shown some success in systematically passing the micro-scale structural mechanics information to the macro-scale hydrodynamics for suspensions with simple polymer conformation and bond potential. The model retains a multi-scaled nature by mapping the polymer configurations into a set of symmetry-preserving macro-scale features. The extended constitutive laws for these macro-scale features can be directly learned from the kinetics of their micro-scale counterparts. In this paper, we develop DeePN2 using more complex micro-structural models. We show that DeePN2 can faithfully capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention. © 2021, CC BY.

关键词： Non Newtonian flow

The mutational constraint spectrum quantified from variation in 141,456 humans (vol 581, pg 434, 2020)

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NATURE 2021年第7874期597卷 E3-E4页

作者： Gudmundsson, Sanna Karczewski, Konrad J. Francioli, Laurent C. Tiao, Grace Cummings, Beryl B. Alfoldi, Jessica Wang, Qingbo Collins, Ryan L. Laricchia, Kristen M. Ganna, Andrea Birnbaum, Daniel P. Gauthier, Laura D. Brand, Harrison Solomonson, Matthew Watts, Nicholas A. Rhodes, Daniel Singer-Berk, Moriel England, Eleina M. Seaby, Eleanor G. Kosmicki, Jack A. Walters, Raymond K. Tashman, Katherine Farjoun, Yossi Banks, Eric Poterba, Timothy Wang, Arcturus Seed, Cotton Whiffin, Nicola Chong, Jessica X. Samocha, Kaitlin E. Pierce-Hoffman, Emma Zappala, Zachary O'Donnell-Luria, Anne H. Minikel, Eric Vallabh Weisburd, Ben Lek, Monkol Ware, James S. Vittal, Christopher Armean, Irina M. Bergelson, Louis Cibulskis, Kristian Connolly, Kristen M. Covarrubias, Miguel Donnelly, Stacey Ferriera, Steven Gabriel, Stacey Gentry, Jeff Gupta, Namrata Jeandet, Thibault Kaplan, Diane Llanwarne, Christopher Munshi, Ruchi Novod, Sam Petrillo, Nikelle Roazen, David Ruano-Rubio, Valentin Saltzman, Andrea Schleicher, Molly Soto, Jose Tibbetts, Kathleen Tolonen, Charlotte Wade, Gordon Talkowski, Michael E. Neale, Benjamin M. Daly, Mark J. MacArthur, Daniel G. Program in Medical and Population Genetics Broad Institute of MIT and Harvard Cambridge MA USA Analytic and Translational Genetics Unit Massachusetts General Hospital Boston MA USA Division of Genetics and Genomics Boston Children’s Hospital Boston MA USA Program in Biological and Biomedical Sciences Harvard Medical School Boston MA USA Program in Bioinformatics and Integrative Genomics Harvard Medical School Boston MA USA Center for Genomic Medicine Massachusetts General Hospital Boston MA USA Institute for Molecular Medicine Finland Helsinki Finland Stanley Center for Psychiatric Research Broad Institute of MIT and Harvard Cambridge MA USA National Heart & Lung Institute and MRC London Institute of Medical Sciences Imperial College London London UK Cardiovascular Research Centre Royal Brompton & Harefield Hospitals NHS Trust London UK Vertex Pharmaceuticals Inc Boston MA USA Department of Pediatrics Harvard Medical School Boston MA USA Broad Genomics Broad Institute of MIT and Harvard Cambridge MA USA Department of Neurology Harvard Medical School Boston MA USA Cardiac Arrhythmia Service and Cardiovascular Research Center Massachusetts General Hospital Boston MA USA Estonian Genome Center Institute of Genomics University of Tartu Tartu Estonia Data Sciences Platform Broad Institute of MIT and Harvard Cambridge MA USA Diabetes Unit Massachusetts General Hospital Boston MA USA Program in Metabolism Broad Institute of MIT and Harvard Cambridge MA USA Department of Medicine Harvard Medical School Boston MA USA Division of General Internal Medicine Massachusetts General Hospital Boston MA USA Department of Public Health Faculty of Medicine University of Helsinki Helsinki Finland Cardiovascular Disease Initiative and Program in Medical and Population Genetics Broad Institute of MIT and Harvard Cambridge MA USA Genomics Diabetes and Endocrinology Lund University Lund Sweden Folkhälsan Institute of Genetics Folkhälsan Research C

The mutational constraint spectrum quantified from variation in 141,456 humans (vol 581, pg 434, 2020)

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NATURE 2021年第7846期590卷 E53-E53页

作者： Karczewski, Konrad J. Francioli, Laurent C. Tiao, Grace Cummings, Beryl B. Alfoldi, Jessica Wang, Qingbo Collins, Ryan L. Laricchia, Kristen M. Ganna, Andrea Birnbaum, Daniel P. Gauthier, Laura D. Brand, Harrison Solomonson, Matthew Watts, Nicholas A. Rhodes, Daniel Singer-Berk, Moriel England, Eleina M. Seaby, Eleanor G. Kosmicki, Jack A. Walters, Raymond K. Tashman, Katherine Farjoun, Yossi Banks, Eric Poterba, Timothy Wang, Arcturus Seed, Cotton Whiffin, Nicola Chong, Jessica X. Samocha, Kaitlin E. Pierce-Hoffman, Emma Zappala, Zachary O'Donnell-Luria, Anne H. Minikel, Eric Vallabh Weisburd, Ben Lek, Monkol Ware, James S. Vittal, Christopher Armean, Irina M. Bergelson, Louis Cibulskis, Kristian Connolly, Kristen M. Covarrubias, Miguel Donnelly, Stacey Ferriera, Steven Gabriel, Stacey Gentry, Jeff Gupta, Namrata Jeandet, Thibault Kaplan, Diane Llanwarne, Christopher Munshi, Ruchi Novod, Sam Petrillo, Nikelle Roazen, David Ruano-Rubio, Valentin Saltzman, Andrea Schleicher, Molly Soto, Jose Tibbetts, Kathleen Tolonen, Charlotte Wade, Gordon Talkowski, Michael E. Aggregation, Genome Neale, Benjamin M. Daly, Mark J. MacArthur, Daniel G. Program in Medical and Population Genetics Broad Institute of MIT and Harvard Cambridge MA USA Analytic and Translational Genetics Unit Massachusetts General Hospital Boston MA USA Program in Biological and Biomedical Sciences Harvard Medical School Boston MA USA Program in Bioinformatics and Integrative Genomics Harvard Medical School Boston MA USA Center for Genomic Medicine Massachusetts General Hospital Boston MA USA Institute for Molecular Medicine Finland Helsinki Finland Stanley Center for Psychiatric Research Broad Institute of MIT and Harvard Cambridge MA USA National Heart & Lung Institute and MRC London Institute of Medical Sciences Imperial College London London UK Cardiovascular Research Centre Royal Brompton & Harefield Hospitals NHS Trust London UK Vertex Pharmaceuticals Inc Boston MA USA Division of Genetics and Genomics Boston Children’s Hospital Boston MA USA Department of Pediatrics Harvard Medical School Boston MA USA Broad Genomics Broad Institute of MIT and Harvard Cambridge MA USA Department of Neurology Harvard Medical School Boston MA USA Cardiac Arrhythmia Service and Cardiovascular Research Center Massachusetts General Hospital Boston MA USA Estonian Genome Center Institute of Genomics University of Tartu Tartu Estonia Data Sciences Platform Broad Institute of MIT and Harvard Cambridge MA USA Diabetes Unit Massachusetts General Hospital Boston MA USA Program in Metabolism Broad Institute of MIT and Harvard Cambridge MA USA Department of Medicine Harvard Medical School Boston MA USA Division of General Internal Medicine Massachusetts General Hospital Boston MA USA Department of Public Health Faculty of Medicine University of Helsinki Helsinki Finland Cardiovascular Disease Initiative and Program in Medical and Population Genetics Broad Institute of MIT and Harvard Cambridge MA USA Genomics Diabetes and Endocrinology Lund University Lund Sweden Folkhälsan Institute of Genetics Folkhälsan Research C

Evaluating drug targets through human loss-of-function genetic variation (vol 581, pg 459, 2020)

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NATURE 2021年第7846期590卷 E56-E56页

作者： Minikel, Eric Vallabh Karczewski, Konrad J. Martin, Hilary C. Cummings, Beryl B. Whiffin, Nicola Rhodes, Daniel Alfoldi, Jessica Trembath, Richard C. van Heel, David A. Daly, Mark J. Schreiber, Stuart L. MacArthur, Daniel G. Program in Medical and Population Genetics Broad Institute of MIT and Harvard Cambridge MA USA Stanley Center for Psychiatric Research Broad Institute of MIT and Harvard Cambridge MA USA Chemical Biology and Therapeutics Science Program Broad Institute of MIT and Harvard Cambridge MA USA Analytical and Translational Genetics Unit Massachusetts General Hospital Boston MA USA Program in Biological and Biomedical Sciences Harvard Medical School Boston MA USA Henry and Allison McCance Center for Brain Health Massachusetts General Hospital Boston MA USA Department of Neurology Massachusetts General Hospital Boston MA USA Prion Alliance Cambridge MA USA National Heart and Lung Institute and MRC London Institute of Medical Sciences Imperial College London London UK National Heart & Lung Institute and MRC London Institute of Medical Sciences Imperial College London London UK Cardiovascular Research Centre Royal Brompton & Harefield Hospitals NHS Trust London UK European Molecular Biology Laboratory European Bioinformatics Institute Wellcome Genome Campus Hinxton Cambridge UK Center for Genomic Medicine Massachusetts General Hospital Boston MA USA Program in Bioinformatics and Integrative Genomics Harvard Medical School Boston MA USA Broad Genomics Broad Institute of MIT and Harvard Cambridge MA USA Division of Genetics and Genomics Boston Children’s Hospital Boston MA USA Department of Pediatrics Harvard Medical School Boston MA USA Cardiac Arrhythmia Service and Cardiovascular Research Center Massachusetts General Hospital Boston MA USA Estonian Genome Center Institute of Genomics University of Tartu Tartu Estonia Department of Medicine Harvard Medical School Boston MA USA Cardiac Arrhythmia Service Massachusetts General Hospital Boston MA USA Centre for Population Genomics Garvan Institute of Medical Research and UNSW Sydney Sydney Australia Centre for Population Genomics Murdoch Children’s Research Institute Melbourne Australia Divi

Transcript expression-aware annotation improves rare variant interpretation (vol 581, pg 452, 2020)

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NATURE 2021年第7846期590卷 E54-E54页

作者： Cummings, Beryl B. Karczewski, Konrad J. Kosmicki, Jack A. Seaby, Eleanor G. Watts, Nicholas A. Singer-Berk, Moriel Mudge, Jonathan M. Karjalainen, Juha Satterstrom, F. Kyle O'Donnell-Luria, Anne H. Poterba, Timothy Seed, Cotton Solomonson, Matthew Alfoldi, Jessica Daly, Mark J. MacArthur, Daniel G. Program in Medical and Population Genetics Broad Institute of MIT and Harvard Cambridge MA USA Analytical and Translational Genetics Unit Massachusetts General Hospital Boston MA USA Program in Biological and Biomedical Sciences Harvard Medical School Boston MA USA Program in Bioinformatics and Integrative Genomics Harvard Medical School Boston MA USA Genomic Informatics Group University Hospital Southampton Southampton UK Stanley Center for Psychiatric Research Broad Institute of MIT and Harvard Cambridge MA USA Division of Genetics and Genomics Boston Children’s Hospital Boston MA USA Department of Pediatrics Harvard Medical School Boston MA USA European Molecular Biology Laboratory European Bioinformatics Institute Wellcome Genome Campus Hinxton Cambridge UK Center for Genomic Medicine Massachusetts General Hospital Boston MA USA Broad Genomics Broad Institute of MIT and Harvard Cambridge MA USA National Heart & Lung Institute and MRC London Institute of Medical Sciences Imperial College London London UK Cardiovascular Research Centre Royal Brompton & Harefield Hospitals NHS Trust London UK Cardiac Arrhythmia Service and Cardiovascular Research Center Massachusetts General Hospital Boston MA USA Estonian Genome Center Institute of Genomics University of Tartu Tartu Estonia Department of Medicine Harvard Medical School Boston MA USA Cardiac Arrhythmia Service Massachusetts General Hospital Boston MA USA Division of General Internal Medicine Massachusetts General Hospital Boston MA USA Institute for Molecular Medicine FIMM University of Helsinki Helsinki Finland Department of Public Health Faculty of Medicine University of Helsinki Helsinki Finland Department of Genetics Harvard Medical School Boston MA USA Data Sciences Platform Broad Institute of MIT and Harvard Cambridge MA USA Genomics Platform Broad Institute of MIT and Harvard Cambridge MA USA Wellcome Sanger Institute Hinxton Cambridgeshire UK Unidad de Investigaci