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Physics-informed machine learning with smoothed particle hydrodynamics: Hierarchy of reduced Lagrangian models of turbulence

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Physical Review Fluids 2023年第5期8卷 054602-054602页

作者： Michael Woodward Yifeng Tian Criston Hyett Chris Fryer Mikhail Stepanov Daniel Livescu Michael Chertkov Graduate Interdisciplinary Program in Applied Mathematics University of Arizona Tucson Arizona 85721 USA Department of Mathematics University of Arizona Tucson Arizona 85721 USA Computer Computational and Statistical Sciences Division LANL Los Alamos New Mexico 87544 USA

Building efficient, accurate, and generalizable reduced-order models of developed turbulence remains a major challenge. This manuscript approaches this problem by developing a hierarchy of parameterized reduced Lagrangian models for turbulent flows, and it investigates the effects of enforcing physical structure through smoothed particle hydrodynamics (SPH) versus relying on neural networks (NNs) as universal function approximators. Starting from NN parametrizations of a Lagrangian acceleration operator, this hierarchy of models gradually incorporates a weakly compressible and parameterized SPH framework, which enforces physical symmetries, such as Galilean, rotational, and translational invariances. Within this hierarchy, two new parameterized smoothing kernels are developed to increase the flexibility of the learn-able SPH simulators. For each model we experiment with different loss functions which are minimized using gradient based optimization, where efficient computations of gradients are obtained by using automatic differentiation and sensitivity analysis. Each model within the hierarchy is trained on two data sets associated with weakly compressible homogeneous isotropic turbulence: (1) a validation set using weakly compressible SPH; and (2) a high-fidelity set from direct numerical simulations. Numerical evidence shows that encoding more SPH structure improves generalizability to different turbulent Mach numbers and time shifts, and that including the novel parameterized smoothing kernels improves the accuracy of SPH at the resolved scales.

关键词： Sensitivity analysis

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Engaging Engineering mathematics students online: Tutoring using MS Teams

Engaging Engineering Mathematics students online: Tutoring u...

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2022 IEEE IFEES World Engineering Education Forum - Global Engineering Deans Council, WEEF-GEDC 2022

作者： Padayachee, Pragashni Campbell, Anita Fredericks, Ebrahim Academic Support Program in Engineering University of Cape Town Cape Town South Africa Centre for Research in Engineering Education University of Cape Town Cape Town South Africa Mathematics University of Cape Town Department of Mathematics and Applied Cape Town South Africa

ISBN: (纸本)9781665475280

Uncommon in the South African higher education landscape, online learning came to the fore during the global pandemic. We present an account of the use of Microsoft Teams for hybrid mathematics tutorials in a one-semester Vector Calculus course at a South African university in 2022. Interviews with the lecturer, analysed through a Community of Inquiry lens, showed the lecturer's perspective of the design and experience for tutors and students. Our aim is to improve the design of future hybrid tutorials to ensure engineering students' capacity in mathematics is well developed, their communication skills are improved, and that they experience working in a team. Future research will evaluate students' and tutors' accounts of their experiences. Our findings raise awareness of the possibilities and potential difficulties when using Microsoft Teams for communication, teaching and learning mathematics. © 2022 IEEE.

关键词： Students

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Neural network representation of electronic structure from ab initio molecular dynamics

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Science Bulletin 2022年第1期67卷 29-37,M0003页

作者： Qiangqiang Gu Linfeng Zhang Ji Feng International Center for Quantum Materials School of PhysicsPeking UniversityBeijing 100871China Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Collaborative Innovation Center of Quantum Matter Beijing 100871China

Despite their rich information content,electronic structure data amassed at high volumes in ab initio molecular dynamics simulations are generally *** introduce a transferable high-fidelity neural network representation of such data in the form of tight-binding Hamiltonians for crystalline *** predictive representation of ab initio electronic structure,combined with machinelearning boosted molecular dynamics,enables efficient and accurate electronic evolution and *** it is applied to a one-dimension charge-density wave material,carbyne,we are able to compute the spectral function and optical conductivity in the canonical *** spectral functions evaluated during soliton-antisoliton pair annihilation process reveal significant renormalization of low-energy edge modes due to retarded electron-lattice coupling beyond the Born-Oppenheimer *** availability of an efficient and reusable surrogate model for the electronic structure dynamical system will enable calculating many interesting physical properties,paving the way to previously inaccessible or challenging avenues in materials modeling.

关键词： Neural network Tight-binding model Electronic structure ab initio molecular dynamics

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Diffusion curvature for estimating local curvature in high dimensional data 22

Diffusion curvature for estimating local curvature in high d...

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Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Dhananjay Bhaskar Kincaid MacDonald Oluwadamilola Fasina Dawson Thomas Bastian Rieck Ian Adelstein Smita Krishnaswamy Department of Genetics Yale University Department of Mathematics Yale University Applied Mathematics Program Yale University Department of Mathematics Department of Physics Yale University Institute of AI for Health Helmholtz Pioneer Campus Helmholtz Munich Department of Computer Science Department of Genetics Applied Mathematics Program Program for Computational Biology and Bioinformatics Yale University

ISBN: (纸本)9781713871088

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data and on estimating local Hessian matrices of neural network loss landscapes.

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A Stochastic Approximate Expectation-Maximization for Structure Determination Directly from Cryo-EM Micrographs

arXiv

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arXiv 2023年

作者： Kreymer, Shay Singer, Amit Bendory, Tamir The School of Electrical Engineering Tel Aviv University Tel Aviv Israel The Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

A single-particle cryo-electron microscopy (cryoEM) measurement, called a micrograph, consists of multiple two-dimensional tomographic projections of a three-dimensional molecular structure at unknown locations, taken under unknown viewing directions. All existing cryo-EM algorithmic pipelines first locate and extract the projection images, and then reconstruct the structure from the extracted images. However, if the molecular structure is small, the signal-to-noise ratio (SNR) of the data is very low, and thus accurate detection of projection images within the micrograph is challenging. Consequently, all standard techniques fail in low-SNR regimes. To recover molecular structures from measurements of low SNR, and in particular small molecular structures, we devise a stochastic approximate expectation-maximization algorithm to estimate the three-dimensional structure directly from the micrograph, bypassing locating the projection images. We corroborate our computational scheme with numerical experiments, and present successful structure recoveries from simulated noisy measurements. © 2023, CC BY.

关键词： Molecular structure

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Revealing Actual Viscoelastic Relaxation Times in Capillary Breakup

arXiv

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arXiv 2025年

作者： Hu, Nan Hwang, Jonghyun Ruangkriengsin, Tachin Stone, Howard A. Department of Mechanical and Aerospace Engineering Princeton University NJ08544 United States Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We use experiments and theory to elucidate the size effect in capillary breakup rheometry, where pre-stretching in the visco-capillary stage causes the apparent relaxation time to be consistently smaller than the actual value. We propose a method accounting for both the experimental size and the finite extensibility of polymers to extract the actual relaxation time. A phase diagram characterizes the expected measurement variability and delineates scaling law conditions. The results refine capillary breakup rheometry for viscoelastic fluids and advance the understanding of breakup dynamics across scales. Copyright © 2025, The Authors. All rights reserved.

关键词： Relaxation time

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SUBSPACE METHOD OF MOMENTS FOR AB INITIO 3-D SINGLE-PARTICLE CRYO-EM RECONSTRUCTION

arXiv

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arXiv 2024年

作者： Hoskins, Jeremy Khoo, Yuehaw Mickelin, Oscar Singer, Amit Wang, Yuguan Departments of Statistics and CCAM University of Chicago Chicago IL 60637 USA Departments of Statistics and CCAM University of Chicago Chicago IL 60637 USA Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Program in Applied and Computational Mathematics and Department of Mathematics Princeton University Princeton NJ 08544 USA Departments of Statistics University of Chicago Chicago IL 60637 USA

Cryo-electron microscopy (Cryo-EM) is a widely-used technique for recovering the 3-D structure of biological molecules from a large number of experimentally generated noisy 2-D tomographic projection images of the 3-D structure, taken from unknown viewing angles. Through computationally intensive algorithms, these observed images are processed to reconstruct the 3-D structures. Many popular computational methods rely on estimating the unknown angles as part of the reconstruction process, which becomes particularly challenging at low signal-to-noise ratio. The method of moments (MoM) offers an alternative approach that circumvents the estimation of viewing angles of individual projection images by instead estimating the underlying distribution of the viewing angles, and is robust to noise given sufficiently many images. However, the method of moments typically entails computing high-order moments of the projection images, incurring significant storage and computational costs. To mitigate this, we propose a new approach called the subspace method of moments (subspace MoM), which compresses the first three moments using data-driven low-rank tensor techniques as well as expansion into a suitable function basis. The compressed moments can be efficiently computed from the set of projection images using numerical quadrature and can be employed to jointly recover the 3-D structure and the distribution of viewing angles. We illustrate the practical applicability of the subspace MoM in numerical experiments using up to the third-order moment, which significantly improves the resolution of MoM reconstructions compared to previous approaches. Copyright © 2024, The Authors. All rights reserved.

关键词： Image reconstruction

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Ab-initio Contrast Estimation and Denoising of Cryo-EM Images

arXiv

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arXiv 2022年

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

Background and Objective: The contrast of cryo-EM images varies from one to another, primarily due to the uneven thickness of the ice layer. This contrast variation can affect the quality of 2-D class averaging, 3-D ab-initio modeling, and 3-D heterogeneity analysis. Contrast estimation is currently performed during 3-D iterative refinement. As a result, the estimates are not available at the earlier computational stages of class averaging and ab-initio modeling. This paper aims to solve the contrast estimation problem directly from the picked particle images in the ab-initio stage, without estimating the 3-D volume, image rotations, or class averages. Methods: The key observation underlying our analysis is that the 2-D covariance matrix of the raw images is related to the covariance of the underlying clean images, the noise variance, and the contrast variability between images. We show that the contrast variability can be derived from the 2-D covariance matrix and we apply the existing Covariance Wiener Filtering (CWF) framework to estimate it. We also demonstrate a modification of CWF to estimate the contrast of individual images. Results: Our method improves the contrast estimation by a large margin, compared to the previous CWF method. Its estimation accuracy is often comparable to that of an oracle that knows the ground truth covariance of the clean images. The more accurate contrast estimation also improves the quality of image restoration as demonstrated in both synthetic and experimental datasets. Conclusions: This paper proposes an effective method for contrast estimation directly from noisy images without using any 3-D volume information. It enables contrast correction in the earlier stage of single particle analysis, and may improve the accuracy of downstream *** Codes 15A29, 65D18, 62H35, 94A08 Copyright © 2022, The Authors. All rights reserved.

关键词： Covariance matrix

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Trustworthy Actionable Perturbations

arXiv

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arXiv 2024年

作者： Friedbaum, Jesse Adiga, Sudarshan Tandon, Ravi Program in Applied Mathematics University of Arizona TucsonAZ United States Department of Electrical and Computer Engineering University of Arizona TucsonAZ United States

Counterfactuals, or modified inputs that lead to a different outcome, are an important tool for understanding the logic used by machine learning classifiers and how to change an undesirable classification. Even if a counterfactual changes a classifier’s decision, however, it may not affect the true underlying class probabilities, i.e. the counterfactual may act like an adversarial attack and "fool" the classifier. We propose a new framework for creating modified inputs that change the true underlying probabilities in a beneficial way which we call Trustworthy Actionable Perturbations (TAP). This includes a novel verification procedure to ensure that TAP change the true class probabilities instead of acting adversarially. Our framework also includes new cost, reward, and goal definitions that are better suited to effectuating change in the real world. We present PAClearnability results for our verification procedure and theoretically analyze our new method for measuring reward. We also develop a methodology for creating TAP and compare our results to those achieved by previous counterfactual methods. © 2024, CC BY.

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Stability of the Inviscid Power-Law Vortex

arXiv

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arXiv 2024年

作者： Binz, Tim Coiculescu, Matei P. Princeton University Program in Applied & Computational Mathematics Fine Hall Washington Road PrincetonNJ08544 United States Princeton University Department of Mathematics Fine Hall Washington Road PrincetonNJ08544 United States

We prove that the power-law vortex ω¯ (x) = β|x|−α, which explicitly solves the stationary unforced incompressible Euler equations in R2 in both physical and self-similar coordinates, is exponentially linearly stable in self-similar coordinates with the natural scaling. This answers a question from [2]. Moreover, we prove that in physical coordinates, the linearization around ω¯ generates a strongly continuous group of isometries and we prove that the spectrum of the linearization is contained in the imaginary axis. Our results are valid for L2(R2) functions with a mild symmetry condition. Copyright © 2024, The Authors. All rights reserved.

关键词： Euler equations

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