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A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics

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Science China mathematics 2020年第7期63卷 1235-1258页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

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 *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.

关键词： two-layer neural network random feature model Gram matrix generalization error

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Solving algebraic riccati equations via proper orthogonal decomposition 19

Solving algebraic riccati equations via proper orthogonal de...

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19th IFAC World Congress on International Federation of Automatic Control, IFAC 2014

作者： Kramer, Boris Department of Mathematics Interdisciplinary Center for Applied Mathematics Virginia Tech. BlacksburgVA24061 United States

ISBN: (纸本)9783902823625

In this paper we present a method to solve algebraic Riccati equations by employing a projection method based on Proper Orthogonal Decomposition. The method only requires simulations of linear systems to compute the solution of a Lyapunov equation. The leading singular vectors are then used to construct a projector which is employed to produce a reduced order system. We compare this approach to an extended Krylov subspace method and a standard Gramian based method. © IFAC.

关键词： Algebra

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Fréchet sensitivity analysis for partial differential equations with distributed parameters

Fréchet sensitivity analysis for partial differential equat...

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作者： Borggaard, Jeff Nunes, Vitor Leite Interdisciplinary Center for Applied Mathematics Department of Mathematics Virginia Tech. United States

ISBN: (纸本)9781457700804

This paper reviews Fréchet sensitivity analysis for partial differential equations with variations in distributed parameters. The Fréchet derivative provides a linear map between parametric variations and the linearized response of the solution. We propose a methodology based on representations of the Fréchet derivative operator to find those variations that lead to the largest changes to the solution (the most significant variations). This includes an algorithm for computing these variations that only requires the action of the Fréchet operator on a given direction (the Gateaux derivative) and its adjoint. This algorithm is applicable since it does not require an approximation of the entire Fréchet operator, but only typical sensitivity analysis software for partial differential equations. The proposed methodology can be utilized to find worst case distributed disturbances and is thus applicable to uncertainty quantification and the optimal placement of sensors and actuators. © 2011 AACC American Automatic Control Council.

关键词： Partial differential equations

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Bifurcation from the essential spectrum for an elliptic equation with general nonlinearities

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Science China mathematics 2023年第10期66卷 2243-2260页

作者： Jianjun Zhang Xuexiu Zhong Huansong Zhou College of Mathematics and Statistics Chongqing Jiaotong UniversityChongqing 400074China South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631China Center for Mathematical Sciences and Department of Mathematics Wuhan University of TechnologyWuhan 430070China

In this paper,based on some prior estimates,we show that the essential spectrum λ=0 is a bifurcation point for a superlinear elliptic equation with only local conditions,which generalizes a series of earlier results ... 详细信息

关键词： elliptic equation bifurcation point essential spectrum

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Laboratory modeling of moist convection using a reactive fluid

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Physical Review Fluids 2025年第5期10卷 053505-053505页

作者： Valentin Dorel Daniel Lecoanet Michael Le Bars CNRS Aix Marseille Université Centrale Marseille IRPHE Marseille France Department of Engineering Sciences and Applied Mathematics and Center for Interdisciplinary Exploration and Research in Astrophysics Northwestern University Evanston Illinois USA

We present a laboratory experiment to model convection in the atmosphere driven by latent heat of condensation (“moist convection”). The system consists of a tank filled with an aqueous solution containing a color indicator that renders the fluid yellow and transparent when acidic and blue and opaque when basic. A sodium lamp positioned above the tank serves as a heat source. Initially, the solution is entirely acidic and stably stratified in temperature. A basic layer is progressively generated at the tank's bottom through water electrolysis, forming an initially stable, blue region. This layer is internally heated by absorbing the heat from the sodium lamp, modeling the latent heat of condensation in the atmosphere. Over time, instabilities develop at the interface between the yellow and blue layers. We investigate the variation of the instability wavelength and complement our observations with a linear stability analysis to elucidate the underlying mechanisms and the process of wavelength selection. Direct numerical simulations are then employed to explore the first steps of the nonlinear regime in connection with experimental observations.

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System-Wide Emergency Policy for Transitioning from Main to Secondary Fuel 63

System-Wide Emergency Policy for Transitioning from Main to ...

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63rd IEEE Conference on Decision and Control, CDC 2024

作者： Pagnier, Laurent Hyett, Criston Ferrando, Robert Goldshtein, Igal Alisse, Jean Saban, Lilah Chertkov, Michael University of Arizona Graduate Interdisciplinary Program in Applied Mathematics and Department of Mathematics TucsonAZ85721 United States Noga the Israel Independent System Operator Haifa Israel

ISBN: (纸本)9798350316339

Faced with the complexities of managing natural gas-dependent power system amid the surge of renewable integration and load unpredictability, this study explores strategies for navigating emergency transitions to costlier secondary fuels. Our aim is to develop decision-support tools for operators during such exigencies. We approach the problem through a Markov Decision Process (MDP) framework, accounting for multiple uncertainties. These include the potential for dual-fuel generator failures and operator response during high-pressure situations. Additionally, we consider the finite reserves of primary fuel, governed by gas-flow partial differential equations (PDEs) and constrained by nodal pressure. Other factors include the variability in power forecasts due to renewable generation and the economic impact of compulsory load shedding. For tractability, we address the MDP in a simplified context, replacing it by Markov Processes evaluated against a selection of policies and scenarios for comparison. Our study considers two models for the natural gas system: an oversimplified model tracking linepack and a more nuanced model that accounts for gas flow network heterogeneity. The efficacy of our methods is demonstrated using a realistic model replicating Israel's power-gas infrastructure. © 2024 IEEE.

关键词： Markov processes

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A Deep Learning Approach for Nerve Injury Classification in Brachial Plexopathies Using Magnetic Resonance Neurography with Modified Hiking Optimization Algorithm

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Academic Radiology 2025年

作者： Dahou, Abdelghani Elaziz, Mohamed Abd Khattap, Mohamed G. Hassan, Hend Galal Eldeen Mohamed Ali School of Computer Science and Technology Zhejiang Normal University Jinhua 321004 China Mathematics and Computer Science department University of Ahmed DRAIA Adrar 01000 Algeria Department of Mathematics Faculty of Science Zagazig University Zagazig 44519 Egypt Faculty of Computer Science and Engineering Galala University Suez 435611 Egypt Artificial Intelligence Research Center (AIRC) Ajman University Ajman 346 United Arab Emirates Department of Diagnostic Interventional Radiology and Molecular Imaging Faculty of Medicine Ain Shams University Cairo 11591 Egypt Technology of Radiology and Medical Imaging Program Faculty of Applied Health Sciences Technology Galala University Suez 435611 Egypt

Rationale and Objectives: Brachial plexopathies (BPs) encompass a complex spectrum of nerve injuries affecting motor and sensory function in the upper extremities. Diagnosis is challenging due to the intricate anatomy and symptom overlap with other neuropathies. Magnetic Resonance Neurography (MRN) provides advanced imaging but requires specialized interpretation. This study proposes an AI-based framework that combines deep learning (DL) with the modified Hiking Optimization Algorithm (MHOA) enhanced by a Comprehensive Learning (CL) technique to improve the classification of nerve injuries (neuropraxia, axonotmesis, neurotmesis) using MRN data. Materials and Methods: The framework utilizes MobileNetV4 for feature extraction and MHOA for optimized feature selection across different MRI sequences (STIR, T2, T1, and DWI). A dataset of 39 patients diagnosed with BP was used. The framework classifies injuries based on Seddon's criteria, distinguishing between normal and abnormal conditions as well as injury severity. Results: The model achieved excellent performance, with 1.0000 accuracy in distinguishing normal from abnormal conditions using STIR and T2 sequences. For injury severity classification, accuracy was 0.9820 in STIR, outperforming the original HOA and other metaheuristic algorithms. Additionally, high classification accuracy (0.9667) was observed in DWI. The proposed framework outperformed traditional methods and demonstrated high sensitivity and specificity. Conclusion: The proposed AI-based framework significantly improves the diagnosis of BP by accurately classifying nerve injury types. By integrating DL and optimization techniques, it reduces diagnostic variability, making it a valuable tool for clinical settings with limited specialized neuroimaging expertise. This framework has the potential to enhance clinical decision-making and optimize patient outcomes through precise and timely diagnoses. © 2025 The Association of University Radiologists

关键词： Brachial Plexopathy (BP) Comprehensive Learning (CL) Deep Learning (DL) Feature Selection (FS) Hiking Optimization Algorithm (HOA) Magnetic Resonance Neurography (MRN) MobileNetV4

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Machine learning from a continuous viewpoint,I

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Science China mathematics 2020年第11期63卷 2233-2266页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

We present a continuous formulation of machine learning,as a problem in the calculus of variations and differential-integral equations,in the spirit of classical numerical *** demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous *** also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous *** discuss how the issues of generalization error and implicit regularization can be studied under this framework.

关键词： machine learning continuous formulation flow-based model gradient flow particle approximation

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Microfluidic droplet detection using synthetic data generated by a Generative Adversarial Network

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Engineering Applications of Artificial Intelligence 2025年 156卷

作者： Karimzadeh, Marziyeh Meybodi, Mohsen Alambardar Shams, Mohammadreza Goli-Malekabadi, Zahra Department of Applied Mathematics and Computer Science Faculty of Mathematics and Statistics University of Isfahan Isfahan81746-73441 Iran Department of Computer Engineering Shahreza Campus University of Isfahan Isfahan Iran Department of Medical Technology and Tissue Engineering School of Life science Engineering College of Interdisciplinary Science and Technology University of Tehran Tehran Iran

Cellular heterogeneity, even among genetically identical cells, results in variations in their properties and behaviors, making single-cell analysis crucial for obtaining detailed insights. However, isolating single cells from a cell population poses major challenges, as conventional laboratory techniques often risk cell damage and involve complex procedures. Droplet microfluidics has emerged as a promising approach for encapsulating cells, particularly single cells, into individual droplets without causing harm. Despite this, factors like cell sedimentation and aggregation can reduce encapsulation efficiency and lead to deviations from the expected Poisson distribution. To address these challenges, leveraging artificial intelligence and deep learning to monitor, detect, and regulate encapsulation conditions in real-time is critical for enhancing system performance. However, deep learning models require substantial training data, and issues like microfluidic channel clogging and the scarcity of certain cell types often limit data availability. To overcome this limitation, researchers are turning to synthetic data generation to supplement training datasets and address data scarcity challenges effectively. This study emphasizes the potential of integrating synthetic data with cutting-edge deep learning techniques to enhance the accuracy and efficiency of single-cell analysis within droplet microfluidic systems. A diverse dataset integrating synthetic and real images was used to train the YOLOv8s model for automated detection and classification of microfluidic droplets, enhancing accuracy and system performance. The model trained on a combination of real and synthetic data outperformed the one trained using conventional data augmentation methods, achieving an mAP0.5 of 98% due to the increased diversity of training images. It also demonstrated faster and more stable training. Additionally, the YOLOv8 network, with a detection rate of approximately 2338 droplets per sec

关键词： Deep learning

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Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

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Communications in Computational Physics 2011年第3期9卷 668-687页

作者： Shi Jin Hao Wu Xu Yang Department of Mathematics University of WisconsinMadisonWI 53706USA Department of Mathematical Sciences Tsinghua UniversityBeijing 10084China Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA

A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical *** this paper,we introduce an efficient semi-Eulerian implementation of this *** new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing *** difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these *** yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase *** a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam ***,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this *** numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

关键词： Schrödinger equation semi-Eulerian Gaussian beam method high order methods

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