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arXiv 2024年

作者： Xu, Sheng Balanov, Amnon Bendory, Tamir Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Electrical Engineering Tel Aviv University Tel Aviv69978 Israel

Accurate orientation estimation is a crucial component of 3D molecular structure reconstruction, both in single-particle cryo-electron microscopy (cryo-EM) and in the increasingly popular field of cryo-electron tomography (cryo-ET). The dominant method, which involves searching for an orientation with maximum cross-correlation relative to given templates, falls short, particularly in low signal-to-noise environments. In this work, we propose a Bayesian framework to develop a more accurate and flexible orientation estimation approach, with the minimum mean square error (MMSE) estimator as a key example. This method effectively accommodates varying structural conformations and arbitrary rotational distributions. Through simulations, we demonstrate that our estimator consistently outperforms the cross-correlation-based method, especially in challenging conditions with low signal-to-noise ratios, and offer a theoretical framework to support these improvements. We further show that integrating our estimator into the iterative refinement in the 3D reconstruction pipeline markedly enhances overall accuracy, revealing substantial benefits across the algorithmic workflow. Finally, we show empirically that the proposed Bayesian approach enhances robustness against the "Einstein from Noise" phenomenon, reducing model bias and improving reconstruction reliability. These findings indicate that the proposed Bayesian framework could substantially advance cryo-EM and cryo-ET by enhancing the accuracy, robustness, and reliability of 3D molecular structure reconstruction, thereby facilitating deeper insights into complex biological systems. © 2024, CC BY.

关键词： Electron microscopy

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Emergent spaces for coupled oscillators

arXiv

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arXiv 2020年

作者： Thiem, Thomas N. Kooshkbaghi, Mahdi Bertalan, Tom Laing, Carlo R. Kevrekidis, Ioannis G. Department of Chemical and Biological Engineering Princeton University United States Program in Applied and Computational Mathematics Princeton University United States Department of Mechanical Engineering Massachusetts Institute of Technology United States School of Natural and Computational Sciences Massey University New Zealand Department of Applied Mathematics and Statistics Johns Hopkins University United States

Systems of coupled dynamical units (e.g. oscillators or neurons) are known to exhibit complex, emergent behaviors that may be simplified through coarse-graining: a process in which one discovers coarse variables and derives equations for their evolution. Such coarse-graining procedures often require extensive experience and/or a deep understanding of the system dynamics. In this paper we present a systematic, data-driven approach to discovering "bespoke" coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical time integrators (initial value solvers). This approach allows us to learn accurate approximations of time derivatives of state variables from sparse flow data, and hence discover useful approximate differential equation descriptions of their dynamic behavior. We demonstrate this capability by learning ODEs that agree with the known analytical expression for the Kuramoto order parameter dynamics at the continuum limit. We then show how this approach can also be used to learn the dynamics of coarse variables discovered through our manifold learning methodology. In both of these examples, we compare the results of our neural network based method to typical finite differences complemented with geometric harmonics. Finally, we present a series of computational examples illustrating how a variation of our manifold learning methodology can be used to discover sets of "effective" parameters, reduced parameter combinations, for multi-parameter models with complex coupling. We conclude with a discussion of possible ext

关键词： Neural networks

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Pre-processing and classification of hyperspectral imagery via selective inpainting

Pre-processing and classification of hyperspectral imagery v...

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IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Victoria Chayes Kevin Miller Rasika Bhalerao Jiajie Luo Wei Zhu Andrea L. Bertozzi Wenzhi Liao Stanley Osher California Research Training Program in Computational and Applied Mathematics United States of America University of California Los Angeles Department of Mathematics United States of America Ghent University Department of Telecommunications and Information Processing Belgium

ISBN: (纸本)9781509041183

We propose a semi-supervised algorithm for processing and classification of hyperspectral imagery. For initialization, we keep 20% of the data intact, and use Principal Component Analysis to discard voxels from noisier bands and pixels. Then, we use either an Accelerated Proximal Gradient algorithm (APGL), or a modified APGL algorithm with a penalty term for distance between inpainted pixels and endmembers (APGL Hyp), on the initialized datacube to inpaint the missing data. APGL and APGL Hyp are distinguished by performance on datasets with full pixels removed or extreme noise. This inpainting technique results in band-by-band datacube sharpening and removal of noise from individual spectral signatures. We can also classify the inpainted cube by assigning each pixel to its nearest endmember via Euclidean distance. We demonstrate improved accuracy in classification over data-mining techniques like k-means, unmixing techniques like Hierarchical Non-Negative Matrix Factorization, and graph-based methods like Non-Local Total Variation.

关键词： Hyperspectral imagery (HSI) data preprocessing image inpainting image classification image enhancement Data preprocessing hyperspectral imagery Image classification image enhancement pixel Pretreatment matrix decomposition initialization Euclidean distance

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Heat transport in liquid water from first-principles and deep neural network simulations

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Physical Review B 2021年第22期104卷 224202-224202页

作者： Davide Tisi Linfeng Zhang Riccardo Bertossa Han Wang Roberto Car Stefano Baroni SISSA–Scuola Internazionale Superiore di Studi Avanzati 34136 Trieste Italy Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing 100088 People's Republic of China Department of Chemistry Department of Physics and Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA CNR Istituto Officina dei Materiali SISSA unit 34136 Trieste Italy

We compute the thermal conductivity of water within linear response theory from equilibrium molecular dynamics simulations, by adopting two different approaches. In one, the potential energy surface (PES) is derived on the fly from the electronic ground state of density functional theory (DFT) and the corresponding analytical expression is used for the energy flux. In the other, the PES is represented by a deep neural network (DNN) trained on DFT data, whereby the PES has an explicit local decomposition and the energy flux takes a particularly simple expression. By virtue of a gauge invariance principle, established by Marcolongo, Umari, and Baroni, the two approaches should be equivalent if the PES were reproduced accurately by the DNN model. We test this hypothesis by calculating the thermal conductivity, at the GGA (PBE) level of theory, using the direct formulation and its DNN proxy, finding that both approaches yield the same conductivity, in excess of the experimental value by approximately 60%. Besides being numerically much more efficient than its direct DFT counterpart, the DNN scheme has the advantage of being easily applicable to more sophisticated DFT approximations, such as meta-GGA and hybrid functionals, for which it would be hard to derive analytically the expression of the energy flux. We find in this way that a DNN model, trained on meta-GGA (SCAN) data, reduces the deviation from experiment of the predicted thermal conductivity by about 50%, leaving the question open as to whether the residual error is due to deficiencies of the functional, to a neglect of nuclear quantum effects in the atomic dynamics, or, likely, to a combination of the two.

关键词： Thermal conductivity Thermal properties Water Density functional theory Linear response theory Machine learning Molecular dynamics

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A Priori Estimates For Two-layer Neural Networks

arXiv

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arXiv 2018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 School of Mathematical Sciences Peking University China

New estimates for the population risk are established for two-layer neural networks. These estimates are nearly optimal in the sense that the error rates scale in the same way as the Monte Carlo error rates. They are equally effective in the over-parametrized regime when the network size is much larger than the size of the dataset. These new estimates are a priori in nature in the sense that the bounds depend only on some norms of the underlying functions to be fitted, not the parameters in the model, in contrast with most existing results which are a posteriori in nature. Using these a priori estimates, we provide a perspective for understanding why two-layer neural networks perform better than kernel *** Codes 41A46, 41A63, 62J02, 65D05 Copyright © 2018, The Authors. All rights reserved.

关键词： Network layers

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Parsimonious reconstruction of network evolution

Parsimonious reconstruction of network evolution

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Lecture Notes in Computer Science

作者： Patro, Rob Sefer, Emre Malin, Justin Marçais, Guillaume Navlakha, Saket Kingsford, Carl Center for Bioinformatics and Computational Biology University of Maryland College Park MD 20742 United States Department of Computer Science University of Maryland College Park MD 20742 United States Computational Biology Bioinformatics and Genomics Concentration Biological Sciences Graduate Program University of Maryland College Park MD 20742 United States Program in Applied Mathematics Statistics and Scientific Computation University of Maryland College Park MD 20742 United States School of Computer Science Carnegie Mellon University Pittsburgh PA 15213 United States

ISBN: (纸本)9783642230370

We consider the problem of reconstructing a maximally parsimonious history of network evolution under models that support gene duplication and loss and independent interaction gain and loss. We introduce a combinatorial framework for encoding network histories, and we give a fast procedure that, given a set of duplication histories, in practice finds network histories with close to the minimum number of interaction gain or loss events. In contrast to previous studies, our method does not require knowing the relative ordering of unrelated duplication events. Results on simulated histories suggest that common ancestral networks can be accurately reconstructed using this parsimony approach. © 2011 Springer-Verlag.

关键词： Genes

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Geometry based data generation 18

Geometry based data generation

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Proceedings of the 32nd International Conference on Neural Information Processing Systems

作者： Ofir Lindenbaum Jay S. Stanley, III Guy Wolf Smita Krishnaswamy Applied Mathematics Program Yale University New Haven CT Computational Biology & Bioinformatics Program Yale University New Haven CT Departments of Genetics & Computer Science Yale University New Haven CT

We propose a new type of generative model for high-dimensional data that learns a manifold geometry of the data, rather than density, and can generate points evenly along this manifold. This is in contrast to existing generative models that represent data density, and are strongly affected by noise and other artifacts of data collection. We demonstrate how this approach corrects sampling biases and artifacts, thus improves several downstream data analysis tasks, such as clustering and classification. Finally, we demonstrate that this approach is especially useful in biology where, despite the advent of single-cell technologies, rare subpopulations and gene-interaction relationships are affected by biased sampling. We show that SUGAR can generate hypothetical populations, and it is able to reveal intrinsic patterns and mutual-information relationships between genes on a single-cell RNA sequencing dataset of hematopoiesis.

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Optimized Interactions for Targeted Self-Assembly: Application to a Honeycomb Lattice

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Physical Review Letters 2005年第22期95卷 228301-228301页

作者： Mikael C. Rechtsman Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics and PRISM Princeton University Princeton New Jersey 08544 USA

We devise an inverse statistical-mechanical methodology to find optimized interaction potentials that lead spontaneously to a target many-particle configuration. Target structures can possess varying degrees of disorder, thus extending the traditional idea of self-assembly to incorporate both amorphous and crystalline structures as well as quasicrystals. For illustration purposes, our computational technique is applied to yield an optimized isotropic (nondirectional) pair potential that spontaneously yields the three-coordinated honeycomb lattice as the ground state structure in two dimensions. This target choice is motivated by its three-dimensional analog, the diamond lattice, which is known to possess desirable photonic band gap properties.

关键词： .honeycomb inverse PRL week ending pair potential spontaneously diamond lattice

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Angular-Momentum-Dependent Orbital-Free Density Functional Theory

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Physical Review Letters 2013年第6期111卷 066402-066402页

作者： Youqi Ke Florian Libisch Junchao Xia Lin-Wang Wang Emily A. Carter Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544 USA Material Science Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA

Orbital-free (OF) density functional theory (DFT) directly solves for the electron density rather than the wave function of many electron systems, greatly simplifying and enabling large scale first principles simulations. However, the required approximate noninteracting kinetic energy density functionals and local electron-ion pseudopotentials severely restrict the general applicability of conventional OFDFT. Here, we present a new generation of OFDFT called angular-momentum-dependent (AMD)-OFDFT to harness the accuracy of Kohn-Sham DFT and the simplicity of OFDFT. The angular momenta of electrons are explicitly introduced within atom-centered spheres so that the important ionic core region can be accurately described. In addition to conventional OF total energy functionals, we introduce a crucial nonlocal energy term with a set of AMD energies to correct errors due to the kinetic energy density functional and the local pseudopotential. We find that our AMD-OFDFT formalism offers substantial improvements over conventional OFDFT, as we show for various properties of the transition metal titanium.

关键词： DENSITY functional theory ELECTRON density KINETIC energy PSEUDOPOTENTIAL method TRANSITION metal compounds ELECTRON-ion collisions

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Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator

arXiv

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arXiv 2017年

作者： Li, Qianxiao Dietrich, Felix Bollt, Erik M. Kevrekidis, Ioannis G. Institute of High Performance Computing Singapore Singapore Faculty of Mathematics Technical University of Munich Germany Department of Mathematics Clarkson University United States Department of Chemical and Biological Engineering Program in Applied and Computational Mathematics Princeton University United States

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables that spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky PDE as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems. Copyright © 2017, The Authors. All rights reserved.

关键词： Machine learning

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