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The barron space and the flow-induced function spaces for neural network models

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arXiv 2019年

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

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: The two-layer networks and the residual neural networks. We define the Barron space and show that it is the right space for two-layer neural network models in the sense that optimal direct and inverse approximation theorems hold for functions in the Barron space. For residual neural network models, we construct the so-called flow-induced function space, and prove direct and inverse approximation theorems for this space. In addition, we show that the Rademacher complexity for bounded sets under these norms has the optimal upper bounds. Copyright © 2019, The Authors. All rights reserved.

关键词： Functional analysis

International Workshop on Applications of Probability and Statistics to Biology,July 11-13,2019--In Honor of Professor Minping Qian’s 80th Birthday

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Quantitative Biology 2020年第2期8卷 177-186页

作者： Minghua Deng Jianfeng Feng Hong Qian Lin Wan Fengzhu Sun Center for Quantitative Biology Peking UniversityBeijing 100871China LMAM Center for Statistical ScienceSchool of Mathematical SciencesPeking UniversityBeijing 100871China Institute of Science and Technology for Brain-inspired Intelligence Fudan UniversityShanghai 200433China School of Mathematical Sciences Fudan UniversityShanghai 200433China Centre for Computational Systems Biology Fudan UniversityShanghai 200433China Key Laboratory of Computational Neuroscience and Brain-Inspired Intelligence at Fudan University Ministry of EducationShanghai 200433China Department of Computer Science University of WarwickCoventryCV47ALUnited Kingdom Department of Applied Mathematics University of WashingtonSeattleWA 98195USA NCMIS Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China Quantitative and Computational Biology Program University of Southern CaliforniaLos AngelesCA 90089USA

The International Workshop on Applications of Probability and Statistics to Biology(APSB)was successfully held in Shanghai,China,July 11-13,*** workshop was hosted by the Institute of Science and Technology for Brain-inspired Intelligence(ISTBI)at Fudan University,and in honor of the 80th birthday of *** Qian of Peking *** of the twenty eight speakers were former students or close collaborators of ***;and there were over eighty participants from all over China and United States.

关键词： Probability speakers ping

A deep learning-based ODE solver for chemical kinetics

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arXiv 2020年

作者： Zhang, Tianhan Zhang, Yaoyu Weinan, E. Ju, Yiguang Department of Mechanical and Aerospace Engineering Princeton University PrincetonNJ08544 United States School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Qing Yuan Research Institute Shanghai Jiao Tong University Shanghai200240 China Department of Mathematics Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Developing efficient and accurate algorithms for chemistry integration is a challenging task due to its strong stiffness and high dimensionality. The current work presents a deep learning-based numerical method called DeepCombustion0.0 to solve stiff ordinary differential equation systems. The homogeneous autoignition of DME/air mixture, including 54 species, is adopted as an example to illustrate the validity and accuracy of the algorithm. The training and testing datasets cover a wide range of temperature, pressure, and mixture conditions between 750 – 1200 K, 30 – 50 atm, and equivalence ratio = 0.7 – 1.5. Both the first-stage low-temperature ignition (LTI) and the second-stage high-temperature ignition (HTI) are considered. The methodology highlights the importance of the adaptive data sampling techniques, power transform preprocessing, and binary deep neural network (DNN) design. By using the adaptive random samplings and appropriate power transforms, smooth submanifolds in the state vector phase space are observed, on which two three-layer DNNs can be appropriately trained. The neural networks are end-to-end, which predict temporal gradients of the state vectors directly. The results show that temporal evolutions predicted by DNN agree well with traditional numerical methods in all state vector dimensions, including temperature, pressure, and species concentrations. Besides, the ignition delay time differences are within 1%. At the same time, the CPU time is reduced by more than 20 times and 200 times compared with the HMTS and VODE method, respectively. The current work demonstrates the enormous potential of applying the deep learning algorithm in chemical kinetics and combustion modeling. © 2020, CC BY.

关键词： Numerical methods

Deep neural network for Wannier function centers

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arXiv 2019年

作者： Zhang, Linfeng Chen, Mohan Wu, Xifan Wang, Han Weinan, E. Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States CAPT HEDPS College of Engineering Peking University Beijing100871 China Department of Physics Temple University PhiladelphiaPA19122 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 Beijing Institute of Big Data Research Beijing100871 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

We introduce a deep neural network (DNN) model that assigns the position of the centers of the electronic charge in the snapshots of a molecular dynamics trajectory. The electronic centers are uniquely specified by the unitary transformation that maps the occupied eigenstates onto maximally localized Wannier functions. In combination with deep potential molecular dynamics, a DNN approach to model the atomic potential energy surface at the ab initio density functional level of theory, the scheme makes possible to predict the dielectric properties of insulators for samples and trajectories inaccessible to direct ab initio simulation, without loss of accuracy. As an example, we report calculations of the infrared absorption spectra of light and heavy water at a dispersion inclusive hybrid functional level of theory, finding good agreement with experiment. Extensions to other spectroscopies, like Raman and sum frequency generation, are discussed. Copyright © 2019, The Authors. All rights reserved.

关键词： Quantum chemistry

CellBiAge: Improved single-cell age classification using data binarization

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Cell Reports 2023年第12期42卷 113500页

作者： Yu, Doudou Li, Manlin Linghu, Guanjie Hu, Yihuan Hajdarovic, Kaitlyn H. Wang, An Singh, Ritambhara Webb, Ashley E. Molecular Biology Cell Biology and Biochemistry Graduate Program Brown University Providence 02912 RI United States Data Science Institute Brown University Providence 02912 RI United States Neuroscience Graduate Program Brown University Providence 02912 RI United States Department of Applied Mathematics & Statistics Johns Hopkins University Baltimore 21218 MD United States Department of Computer Science Brown University Providence 02912 RI United States Center for Computational Molecular Biology Brown University Providence 02912 RI United States Department of Molecular Biology Cell Biology and Biochemistry Brown University Providence 02912 RI United States Center on the Biology of Aging Brown University Providence 02912 RI United States Carney Institute for Brain Science Brown University Providence 02912 RI United States Center for Translational Neuroscience Brown University Providence 02912 RI United States

Aging is a major risk factor for many diseases. Accurate methods for predicting age in specific cell types are essential to understand the heterogeneity of aging and to assess rejuvenation strategies. However, classifying organismal age at single-cell resolution using transcriptomics is challenging due to sparsity and noise. Here, we developed CellBiAge, a robust and easy-to-implement machine learning pipeline, to classify the age of single cells in the mouse brain using single-cell transcriptomics. We show that binarization of gene expression values for the top highly variable genes significantly improved test performance across different models, techniques, sexes, and brain regions, with potential age-related genes identified for model prediction. Additionally, we demonstrate CellBiAge's ability to capture exercise-induced rejuvenation in neural stem cells. This study provides a broadly applicable approach for robust classification of organismal age of single cells in the mouse brain, which may aid in understanding the aging process and evaluating rejuvenation methods. © 2023 The Authors

关键词： aging brain CP: Cell biology hypothalamus machine learning single-cell RNA-seq

Overview of the Head and Neck Tumor Segmentation for Magnetic Resonance Guided Applications (HNTS-MRG) 2024 Challenge

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arXiv 2024年

作者： Wahid, Kareem A. Dede, Cem El-Habashy, Dina M. Kamel, Serageldin Rooney, Michael K. Khamis, Yomna Abdelaal, Moamen R.A. Ahmed, Sara Corrigan, Kelsey L. Chang, Enoch Dudzinski, Stephanie O. Salzillo, Travis C. McDonald, Brigid A. Mulder, Samuel L. McCullum, Lucas Alakayleh, Qusai Sjogreen, Carlos He, Renjie Mohamed, Abdallah S.R. Lai, Stephen Y. Christodouleas, John P. Schaefer, Andrew J. Naser, Mohamed A. Fuller, Clifton D. Department of Radiation Oncology The University of Texas MD Anderson Cancer HoustonTX United States Department of Imaging Physics The University of Texas MD Anderson Cancer HoustonTX United States Transitional Year Program Corewell Health Wiliam Beaumont Royal OakMI United States Department of Radiation Oncology University of Maryland School of Medicine BaltimoreMD United States Department of Clinical Oncology and Nuclear Medicine Faculty of Medicine Alexandria University Alexandria Egypt UT MD Anderson Cancer Center UTHealth Houston Graduate School of Biomedical Sciences Houston United States Department of Radiation Oncology Baylor College of Medicine HoustonTX United States Department of Head and Neck Surgery The University of Texas MD Anderson Cancer HoustonTX United States Elekta AtlantaGA United States Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX United States

Magnetic resonance (MR)-guided radiation therapy (RT) is enhancing head and neck cancer (HNC) treatment through superior soft tissue contrast and longitudinal imaging capabilities. However, manual tumor segmentation remains a significant challenge, spurring interest in artificial intelligence (AI)-driven automation. To accelerate innovation in this field, we present the Head and Neck Tumor Segmentation for MR-Guided Applications (HNTS-MRG) 2024 Challenge, a satellite event of the 27th International Conference on Medical Image Computing and Computer Assisted Intervention. This challenge addresses the scarcity of large, publicly available AI-ready adaptive RT datasets in HNC and explores the potential of incorporating multi-timepoint data to enhance RT auto-segmentation performance. Participants tackled two HNC segmentation tasks: automatic delineation of primary gross tumor volume (GTVp) and gross metastatic regional lymph nodes (GTVn) on pre-RT (Task 1) and mid-RT (Task 2) T2-weighted scans. The challenge provided 150 HNC cases for training and 50 for testing, hosted on *** using a Docker submission framework. In total, 19 independent teams from across the world qualified by submitting both their algorithms and corresponding papers, resulting in 18 submissions for Task 1 and 15 submissions for Task 2. Evaluation using the mean aggregated Dice Similarity Coefficient showed top-performing AI methods achieved scores of 0.825 in Task 1 and 0.733 in Task 2. These results surpassed clinician interobserver variability benchmarks, marking significant strides in automated tumor segmentation for MR-guided RT applications in HNC. © 2024, CC BY.

关键词： Magnetic resonance imaging

Superposition and mimicking theorems for conditional McKean-Vlasov equations

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arXiv 2020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 & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Princeton University PrincetonNJ08544 United States

We consider conditional McKean-Vlasov stochastic differential equations (SDEs), such as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs satisfy non-linear stochastic partial differential equations (SPDEs) of the second order, whereas the laws of the conditional time-marginals follow Fokker-Planck equations on the space of probability measures. We prove two superposition principles: The first establishes that any solution of the SPDE can be lifted to a solution of the conditional McKean-Vlasov SDE, and the second guarantees that any solution of the Fokker-Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Itô process can be emulated by those of a solution to a conditional McKean-Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean-Vlasov *** Codes 60H10, 60H15, 35Q84, 93E20 Copyright © 2020, The Authors. All rights reserved.

关键词： Fokker Planck equation

Convergence acceleration of ensemble Kalman inversion in nonlinear settings

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arXiv 2019年

作者： Chada, Neil K. Tong, Xin T. Applied Mathematics and Computational Science Program King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia Department of Mathematics National University of Singapore 119077 Singapore

Many data-science problems can be formulated as an inverse problem, where the parameters are estimated by minimizing a proper loss function. When complicated black-box models are involved, derivative-free optimization tools are often needed. The ensemble Kalman filter (EnKF) is a particle-based derivative-free Bayesian algorithm originally designed for data assimilation. Recently, it has been applied to inverse problems for computational efficiency. The resulting algorithm, known as ensemble Kalman inversion (EKI), involves running an ensemble of particles with EnKF update rules so they can converge to a minimizer. In this article, we investigate EKI convergence in general nonlinear settings. To improve convergence speed and stability, we consider applying EKI with non-constant step-sizes and covariance inflation. We prove that EKI can hit critical points with finite steps in non-convex settings. We further prove that EKI converges to the global minimizer polynomially fast if the loss function is strongly convex. We verify the analysis presented with numerical experiments on two inverse problems. Copyright © 2019, The Authors. All rights reserved.

关键词： Inverse problems

A Mathematical Model for Linguistic Universals

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arXiv 2019年

作者： Weinan, E. Zhou, Yajun Department of Mathematics & Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

Inspired by chemical kinetics and neurobiology, we propose a mathematical theory for pattern recurrence in text documents, applicable to a wide variety of languages. We present a Markov model at the discourse level for Steven Pinker’s "mentalese", or chains of mental states that transcend the spoken/written forms. Such (potentially) universal temporal structures of textual patterns lead us to a language-independent semantic representation, or a translationally-invariant word embedding, thereby forming the common ground for both comprehensibility within a given language and translatability between different languages. Applying our model to documents of moderate lengths, without relying on external knowledge bases, we reconcile Noam Chomsky’s "poverty of stimulus" paradox with statistical learning of natural languages. Copyright © 2019, The Authors. All rights reserved.

关键词： Markov processes