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arXiv 2025年

作者： Shi, Yunpeng Singer, Amit Verbeke, Eric J. Department of Mathematics University of California at Davis DavisCA United States Department of Mathematics Princeton University PrincetonNJ United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States

Many applications of computer vision rely on the alignment of similar but non-identical images. We present a fast algorithm for aligning heterogeneous images based on optimal transport. Our approach combines the speed of fast Fourier methods with the robustness of sliced probability metrics and allows us to efficiently compute the alignment between two L × L images using the sliced 2-Wasserstein distance in O(L2 log L) operations. We show that our method is robust to translations, rotations and deformations in the images. Copyright © 2025, The Authors. All rights reserved.

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Deuteration removes quantum dipolar defects from KDP crystals

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npj computational Materials 2024年第1期10卷 684-693页

作者： Bingjia Yang Pinchen Xie Roberto Car Department of Chemistry Princeton UniversityPrincetonNJ 08544USA Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Department of Chemistry Department of PhysicsProgram in Applied and Computational MathematicsPrinceton Materials InstitutePrinceton UniversityPrincetonNJ 08544USA

Dielectric properties of the hydrogen-bonded ferroelectric crystal KH_(2)PO_(4)(KDP)differ significantly from those of KD_(2)PO_(4)(DKDP).It is well established that deuteration affects the interplay of hydrogenbond switches and heavy ion displacements that underlie the emergence of macroscopic polarization,but a detailed microscopic model is *** show that all-atompath integral molecular dynamics simulations can predict the isotope effects,revealing the microscopic mechanism that differentiates KDP and *** tunneling generates phosphate configurations that do not contribute to the *** low temperatures,these quantum dipolar defects are substantial in KDP but negligible in *** intrinsic defects explain why KDP has lower spontaneous polarization and transition entropy than *** prominent role of quantum fluctuations in KDP is related to the unusual strength of the hydrogen bonds and should be equally important in other crystals of the KDP family,which exhibit similar isotope effects.

关键词： quantum KDP remove

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Structural properties of hyperuniform Voronoi networks

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Physical Review E 2025年第3期111卷 034123-034123页

作者： Eli Newby Wenlong Shi Yang Jiao Reka Albert Salvatore Torquato Department of Physics Pennsylvania State University University Park Pennsylvania 16802 USA Materials Science and Engineering Arizona State University Tempe Arizona 85287 USA Department of Physics Arizona State University Tempe Arizona 85287 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by the complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, while lacking conventional long-range order, as in liquids and glasses. In this work, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform Poisson process, nonhyperuniform random sequential addition (RSA) saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks by determining multiple metrics that characterize the distribution, including their average areas and corresponding variances as well as higher-order cumulants (i.e., skewness γ1 and excess kurtosis γ2). We show that the HIP distribution is far from Gaussian, as evidenced by a high skewness (γ1=3.16) and large positive excess kurtosis (γ2=16.2). The Poisson (with γ1=1.07 and γ2=1.79) and non-stealthy hyperuniform (with γ1=0.257 and γ2=0.0217) distributions are Gaussian-like distributions, since they exhibit a small but positive skewness and excess kurtosis. The RSA (with γ1=0.450 and γ2=−0.0384) and the highest stealthy hyperuniform distributions (with γ1=0.0272 and γ2=−0.0626) are also non-Gaussian because of their low skewness and negative excess kurtosis, which is diametrically opposite of the non-Gaussian behavior of the HIP. The fact that the cell-area distributions of large, finite-sized RSA and stealthy hyperuniform networks (e.g., with N≈10,000 nodes) are narrower, have larger peaks, and smaller tails than a Gaussian distribution implies that in the thermodynamic limit th

关键词： Gaussian distribution

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Optimization-Based String Method for Finding Minimum Energy Path

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Communications in computational Physics 2013年第7期14卷 265-275页

作者： Amit Samanta Weinan E Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Beijing International Center for Mathematical Research Peking UniversityBeijingChina

We present an efficient algorithm for calculating the minimum energy path(MEP)and energy barriers between local minima on a multidimensional potential energy surface(PES).Such paths play a central role in the understanding of transition pathways between metastable *** method relies on the original formulation of the string method[***.B,66,052301(2002)],*** evolve a smooth curve along a direction normal to the *** algorithm works by performing minimization steps on hyperplanes normal to the *** the problem of finding MEP on the PES is remodeled as a set of constrained minimization *** provides the flexibility of using minimization algorithms faster than the steepest descent method used in the simplified string method[***.,126(16),164103(2007)].At the same time,it provides a more direct analog of the finite temperature string *** applicability of the algorithm is demonstrated using various examples.

关键词： Rare events string method minimum energy path dislocation nucleation

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Integrating Chain-of-Thought and Retrieval Augmented Generation Enhances Rare Disease Diagnosis from Clinical Notes

arXiv

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arXiv 2025年

作者： Wu, Da Wang, Zhanliang Nguyen, Quan Wang, Kai Raymond G. Perelman Center for Cellular and Molecular Therapeutics Children's Hospital of Philadelphia PhiladelphiaPA19104 United States Applied Mathematics and Computational Science Graduate Program University of Pennsylvania PhiladelphiaPA19104 United States Bioengineering Graduate Program University of Pennsylvania PhiladelphiaPA19104 United States Department of Pathology and Laboratory Medicine Perelman School of Medicine University of Pennsylvania PhiladelphiaPA19104 United States

Background: Several studies show that large language models (LLMs) struggle with phenotype-driven gene prioritization for rare diseases. These studies typically use Human Phenotype Ontology (HPO) terms to prompt foundation models like GPT and LLaMA to predict candidate genes. However, in real-world settings, foundation models are not optimized for domain-specific tasks like clinical diagnosis, yet inputs are unstructured clinical notes rather than standardized terms. How LLMs can be instructed to predict candidate genes or disease diagnosis from unstructured clinical notes remains a major challenge. Methods: We introduce RAG-driven CoT and CoT-driven RAG, two methods that combine Chain-of-Thought (CoT) and Retrieval Augmented Generation (RAG) to analyze clinical notes. A five-question CoT protocol mimics expert reasoning, while RAG retrieves data from sources like HPO and OMIM (Online Mendelian Inheritance in Man). We evaluated these approaches on rare disease datasets, including 5,980 Phenopacket-derived notes, 255 literature-based narratives, and 220 in-house clinical notes from Children’s Hospital of Philadelphia. Results: We found that recent foundations models, including Llama 3.3-70B-Instruct and DeepSeek-R1-Distill-Llama-70B, outperformed earlier versions such as Llama 2 and GPT-3.5. We also showed that RAG-driven CoT and CoT-driven RAG both outperform foundation models in candidate gene prioritization from clinical notes;in particular, both methods with DeepSeek backbone resulted in a top-10 gene accuracy of over 40% on Phenopacket-derived clinical notes. RAG-driven CoT works better for high-quality notes, where early retrieval can anchor the subsequent reasoning steps in domain-specific evidence, while CoT-driven RAG has advantage when processing lengthy and noisy notes. Conclusions: Integrating CoT and RAG enhances LLMs’ understanding of clinical notes in the context of rare disease diagnosis, and can facilitate various downstream medical tasks. © 2025, CC

关键词： Chains

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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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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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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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Learning algorithms for mean field optimal control

arXiv

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arXiv 2025年

作者： Soner, H. Mete Teichmann, Josef Yan, Qinxin Department of Operations Research and Financial Engineering Princeton University PrincetonNJ08540 United States Department of Mathematics ETH Zürich Switzerland Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

We analyze an algorithm to numerically solve the mean-field optimal control problems by approximating the optimal feedback controls using neural networks with problem specific architectures. We approximate the model by an N-particle system and leverage the exchangeability of the particles to obtain substantial computational efficiency. In addition to several numerical examples, a convergence analysis is provided. We also developed a universal approximation theorem on Wasserstein *** Codes 35Q89, 35D40, 49L25, 60G99 © 2025, CC BY-NC-ND.

关键词： Optimal control systems

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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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