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1st Challenge on Head and Neck Tumor Segmentation for MRI-Guided Applications, HNTS-MRG 2024, Held in Conjunction with 27th International Conference on Medical Image Computing and Computer-Assisted Intervention, MICCAI 2024

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 CenterUTHealth 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

ISBN: (纸本)9783031832734

Magnetic resonance (MR)-guided radiation therapy (RT) is enhanc- ing head and neck cancer (HNC) treatment through superior soft tissue con- trast 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 final 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 Simi- larity 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 vari- ability benchmarks, marking significant strides in automated tumor segmentation for MR-guided RT applications in HNC. © The Author(s) 2025.

关键词： Deep learning

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NONOSCILLATORY SPECTRAL ELEMENT CHEBYSHEV METHOD FOR SHOCK-WAVE CALCULATIONS

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JOURNAL OF computational PHYSICS 1993年第1期107卷 10-22页

作者： SIDILKOVER, D KARNIADAKIS, GE Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

A new algorithm based on spectral element discretization and non-oscillatory ideas is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell averaging and reconstruction procedures, that employs a staggered grid of Gauss-Chebyshev and Gauss-Lobatto Chebyshev discretizations. The non-oscillatory reconstruction procedure is based on ideas similar to those proposed by Cai et al. (Math. Comput. 52, 389 (1989)) but employs a modified technique which is more robust and simpler in terms of determining the location and strength of a discontinuity. It is demonstrated through model problems of linear advection, inviscid Burgers equation, and one-dimensional Euler system that the proposed algorithm leads to stable, non-oscillatory accurate results. Exponential accuracy away from the discontinuity is realized for the inviscid Burgers equation example.

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

IEEE Transactions on Artificial Intelligence

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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

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ANOMALOUS DIMENSIONS IN TURBULENCE

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MODERN PHYSICS LETTERS A 1991年第11期6卷 1023-1043页

作者： MIGDAL, AA Physics Department and Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA

A time-independent field theoretical framework for turbulence is suggested, based upon a variational principle for a stationary solution of the Fokker-Planck equation. We obtain a functional equation for the effective Action of this spatial field theory and investigate its general properties and some numerical solutions. The equation is completely universal, and allows for the scale invariant solutions in the inertial range. The critical indices are not fixed at the kinematical level, but rather should be found from certain eigenvalue conditions, as in the field theory of critical phenomena. Unlike the Wyld field theory, there are no divergences in our Feynman integrals, due to some magic cancellations. The simplest possible Gaussian approximation yields crude but still reasonable results (there are deviations from Kolmogorov scaling in 3 dimensions, but at 2.7544 dimensions it would be exact). Our approach allows us to study some new problems, such as spontaneous parity breaking in 3d turbulence. It turns out that with the appropriate helicity term added to the velocity correlation function, logarithmic infrared divergences arise in our field theory which effectively eliminates these terms. In order to build a quantitative theory of turbulence, one should consider more sophisticated Ansatz for the effective Action, which would require serious numerical work.

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A Qualitative Study of the Dynamic Behavior for Adaptive Gradient Algorithms 2

A Qualitative Study of the Dynamic Behavior for Adaptive Gra...

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2nd Mathematical and Scientific Machine Learning Conference, MSML 2021

作者： Ma, Chao Wu, Lei Weinan, E. Department of Mathematics Stanford University United States Program in Applied and Computational Mathematics Princeton University United States Program in Applied and Computational Mathematics Department of Mathematics Princeton University United States

The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on the choice of the hyper-parameters: oscillations, spikes, and divergence. In particular, Adam converges much smoother and even faster when the values of the two momentum factors are close to each other. This observation is particularly important for scientific computing tasks, for which the training process usually proceeds into the high precision regime. © 2021 C. Ma, L. Wu & W. E.

关键词： Gradient methods

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Simulation of Impurity Diffusion in a Strained Nanowire Using Off-Lattice KMC

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Communications in computational Physics 2007年第1期2卷 164-176页

作者： Weidong Guo Tim P.Schulze Weinan E Department of Mathematics University of TennesseeKnoxvilleTN 37996-1300USA Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544-1000USA.

Kinetic Monte Carlo(KMC)is a stochastic model used to simulate crystal ***,most KMC models rely on a pre-defined lattice that neglects dislocations,lattice mismatch and strain *** this paper,we investigate the use of a 3D off-lattice KMC *** test this method by investigating impurity diffusion in a strained FCC *** faster than a molecular dynamics simulation,the most general implementation of off-lattice KMC is much slower than a lattice-based *** improved procedure is achieved for weakly strained systems by precomputing approximate saddle point locations based on unstrained lattice *** this way,one gives up some of the flexibility of the general method to restore some of the computational speed of lattice-based *** addition to providing an alternative approach to nano-materials simulation,this type of simulation will be useful for testing and calibrating methods that seek to parameterize the variation in the transition rates for lattice-based KMC using continuum modeling.

关键词： Off-lattice KMC nanowire strain rate impurity diffusion.

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MORSE AND MELNIKOV FUNCTIONS FOR NLS PDES

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COMMUNICATIONS IN MATHEMATICAL PHYSICS 1994年第1期162卷 175-214页

作者： LI, Y MCLAUGHLIN, DW 1. Department of Mathematics Princeton University 08544 Princeton NJ USA 2. Department of Mathematics Program in Applied and Computational Mathematics Princeton University 08544 Princeton NJ USA

The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat liner operator L, is developed in sufficient detail for later use in studies of perturbations of the NLS equation. ''Counting lemmas'' for the non-selfadjoint operator L, are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite disc D in the complex eigenvalue plane. The radius of the disc D is controlled by the H-1 norm of the potential q. For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds - ''whiskered tori'' for the NLS pde. The Floquet discriminant DELTA(lambda;q) used to introduce a natural sequence of NLS constants of motion, [F(j)(q) = DELTA(lambda = lambda(j)c(q);q), where lambda(j)c denotes the j(th) critical point of the Floquet discriminant DELTA(lambda)]. A Taylor series expansion of the constants F(j)(q), with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of F(j)(q), which themselves are expressed in terms of quadratic products of eigenfunctions of L. The second variation permits identification, within the disc D, of important bifurcations m the spectral configurations of the operator L. The constant F(j)(q), as the height of the Floquet discriminant over the critical point lambda(j)c, admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, {F(j)(q)}\j\>N, which is associated with critical points outside the disc D. Within this disc, the interpretation is only valid locally, with the s

关键词： 34L05 35Q55 58F05 58F19

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A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrdinger Equations with Multiply Components

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Communications in Theoretical Physics 2014年第6期61卷 703-709页

作者：钱旭宋松和李伟斌 Department of Mathematics and Systems Science National University of Defense Technology Program in Applied and Computational Mathematics Princeton University State Key Laboratory of High Performance Computing National University of Defense Technology

Considering the coupled nonlinear Schr¨odinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time ***, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.

关键词： conservative algorithm coupled nonlinear SchrSdinger equation multiply components unstablewave

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mathematics for cryo-electron microscopy

Mathematics for cryo-electron microscopy

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2018 International Congress of Mathematicians, ICM 2018

作者： Singer, Amit Department of Mathematics Program in Applied and Computational Mathematic Princeton University United States

ISBN: (纸本)9789813272934

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM. © ICM *** rights reserved.

关键词： X ray crystallography

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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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