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arXiv 2023年

作者： Woodward, Michael Tian, Yifeng Mohan, Arvind Lin, Yen Ting Hader, Christoph Fasel, Hermann Chertkov, Misha Livescu, Daniel Graduate Interdisciplinary Program in Applied Mathematics Department of Mathematics University of Arizona TucsonAZ85721 United States Computer Computational and Statistical Sciences Division Los Alamos National Laboratory Los AlamosNM87544 United States Department of Aerospace and Mechanical Engineering University of Arizona TucsonAZ85721 United States

In this work, we apply, for the first time to spatially inhomogeneous flows, a recently developed data-driven learning algorithm of Mori-Zwanzig (MZ) operators, which is based on a generalized Koopman’s description of dynamical systems. The MZ formalism provides a mathematically exact procedure for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. The algorithm developed in this work applies Mori’s linear projection operator and an SVD based compression to the selection of the resolved variables (equivalently, a low rank approximation of the two time covariance matrices). We show that this MZ decomposition not only identifies the same spatio-temporal structures found by DMD, but it can also be used to extract spatio-temporal structures of the hysteresis effects present in the memory kernels. We perform an analysis of these structures in the context of a laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6, and show the dynamical relevance of the memory kernels. Additionally, by including these memory terms learned in our data-driven MZ approach, we show improvement in prediction accuracy over DMD at the same level of truncation and at a similar computational cost. Furthermore, an analysis of the spatio-temporal structures of the MZ operators shows identifiable structures associated with the nonlinear generation of the so-called "hot" streaks on the surface of the flared cone, which have previously been observed in experiments and direct numerical simulations. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamic mode decomposition

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Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems

arXiv

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arXiv 2019年

作者： Luo, Xiaopeng Xu, Xin Rabitz, Herschel Department of Chemistry Princeton University PrincetonNJ08544 United States School of Managment and Engineering Nanjing University Nanjing210008 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

In this paper, a meshless Hermite-HDMR finite difference method is proposed to solve high-dimensional Dirichlet problems. The approach is based on the local Hermite-HDMR expansion with an additional smoothing technique. First, we introduce the HDMR decomposition combined with the multiple Hermite series to construct a class of Hermite-HDMR approximations, and the relevant error estimate is theoretically built in a class of Hermite spaces. It can not only provide high order convergence but also retain good scaling with increasing dimensions. Then the Hermite-HDMR based finite difference method is particularly proposed for solving high-dimensional Dirichlet problems. By applying a smoothing process to the Hermite-HDMR approximations, numerical stability can be guaranteed even with a small number of nodes. Numerical experiments in dimensions up to 30 show that resulting approximations are of very high quality. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite difference method

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Numerical meshless solution of high-dimensional sine-gordon equations via fourier HDMR-HC approximation

arXiv

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arXiv 2019年

作者： Xu, Xin Luo, Xiaopeng Rabitz, Herschel Department of Chemistry Princeton University PrincetonNJ08544 United States School of Managment and Engineering Nanjing University Nanjing210008 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs. Copyright © 2019, The Authors. All rights reserved.

关键词： Numerical methods

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Deformations of acid-mediated invasive tumors in a model with Allee effect

arXiv

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arXiv 2024年

作者： Carter, Paul Doelman, Arjen van Heijster, Peter Levy, Daniel Maini, Philip Okey, Erin Yeung, Paige Department of Mathematics University of California Irvine United States Mathematical Institute Leiden University Leiden Netherlands Mathematical and Statistical Methods—Biometris Wageningen University & Research Wageningen Netherlands Program in Applied and Computational Mathematics Princeton University Princeton United States Mathematical Institute University of Oxford Oxford United Kingdom Division of Applied Mathematics Brown University ProvidenceRI United States Massachusetts Institute of Technology Cambridge United States

We consider a Gatenby–Gawlinski-type model of invasive tumors in the presence of an Allee effect. We describe the construction of bistable one-dimensional traveling fronts using singular perturbation techniques in different parameter regimes corresponding to tumor interfaces with, or without, acellular gap. By extending the front as a planar interface, we perform a stability analysis to long wavelength perturbations transverse to the direction of front propagation and derive a simple stability criterion for the front in two spatial dimensions. In particular we find that in general the presence of the acellular gap indicates transversal instability of the associated planar front, which can lead to complex interfacial dynamics such as the development of finger-like protrusions and/or different invasion speeds. © 2024, CC BY.

关键词： Perturbation techniques

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Applying Graph Theory and Mathematical-computational Modelling to Study a Neurophysiological Circuit

引用

Open Journal of Modelling and Simulation 2021年第2期9卷 159-171页

作者： Camila de Andrade Kalil Maria Clícia Stelling de Castro Dilson Silva Célia Martins Cortez Postgraduate Program in Computational Sciences Rio de Janeiro State University Rio de Janeiro Brazil Department of Informatics and Computer Science Rio de Janeiro State University Rio de Janeiro Brazil Department of Applied Mathematics Rio de Janeiro State University Rio de Janeiro Brazil

The aim of the present study is to contribute to the knowledge about the functioning of the neuronal circuits. We built a mathematical-computational model using graph theory for a complex neurophysiological circuit consisting of a reverberating neuronal circuit and a parallel neuronal circuit, which could be coupled. Implementing our model in C++ and applying neurophysiological values found in the literature, we studied the discharge pattern of the reverberant circuit and the parallel circuit separately for the same input signal pattern, examining the influence of the refractory period and the synaptic delay on the respective output signal patterns. Then, the same study was performed for the complete circuit, in which the two circuits were coupled, and the parallel circuit could then influence the functioning of the reverberant. The results showed that the refractory period played an important role in forming the pattern of the output spectrum of a reverberating circuit. The inhibitory action of the parallel circuit was able to regulate the reverberation frequency, suggesting that parallel circuits may be involved in the control of reverberation circuits related to motive activities underlying precision tasks and perhaps underlying neural work processes and immediate memories.

关键词： Mathematical-computational Modelling Neurophysiological Circuit Reverberating Circuit Parallel Circuit

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CVIT: CONTINUOUS VISION TRANSFORMER FOR OPERATOR LEARNING

arXiv

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arXiv 2024年

作者： Wang, Sifan Seidman, Jacob H. Sankaran, Shyam Wang, Hanwen Pappas, George J. Perdikaris, Paris Institution for Foundation of Data Science Yale University United States Graduate Program in Applied Mathematics and Computational Science University of Pennsylvania United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania United States Department of Electrical and Systems Engineering University of Pennsylvania United States

Operator learning, which aims to approximate maps between infinite-dimensional function spaces, is an important area in scientific machine learning with applications across various physical domains. Here we introduce the Continuous Vision Transformer (CViT), a novel neural operator architecture that leverages advances in computer vision to address challenges in learning complex physical systems. CViT combines a vision transformer encoder, a novel grid-based coordinate embedding, and a query-wise cross-attention mechanism to effectively capture multi-scale dependencies. This design allows for flexible output representations and consistent evaluation at arbitrary resolutions. We demonstrate CViT’s effectiveness across a diverse range of partial differential equation (PDE) systems, including fluid dynamics, climate modeling, and reaction-diffusion processes. Our comprehensive experiments show that CViT achieves state-of-the-art performance on multiple benchmarks, often surpassing larger foundation models, even without extensive pretraining and roll-out fine-tuning. Taken together, CViT exhibits robust handling of discontinuous solutions, multi-scale features, and intricate spatio-temporal dynamics. Our contributions can be viewed as a significant step towards adapting advanced computer vision architectures for building more flexible and accurate machine learning models in the physical sciences. All data and code are publicly available at https://***/PredictiveIntelligenceLab/cvit. © 2024, CC BY-NC-SA.

关键词： Computer vision

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Orthogonal die random measures, primes, and applications

arXiv

引用

arXiv 2020年

作者： Bastian, Caleb Deen Rempala, Grzegorz A. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Massive Dynamics PrincetonNJ08542 United States Department of Mathematics Ohio State University ColumbusOH43210 United States Division of Biostatistics Ohio State University ColumbusOH43210 United States

We show that the sequence of thinned uniform random counting measures converges weakly to the Poisson random measure. We call such measures ‘orthogonal dice’ due to their natural connection with certain counting problems and link the sizes of orthogonal dice with the primes by constructing the largest known prime orthogonal die. We give many examples of possible use of the construct of orthogonal dice in various areas of applications including gambling, cosmology, random matrices, approximation theory and circuits. The gambling example reveals for instance that fair six-sided die numbered {1, 2, 3, 4, 5, 6} generate negative covariance in point representations of hands across players, in contrast to seven-sided dice numbered {1, 2, 3, 4, 5, 6, 7} that generate zero covariance and are orthogonal. The cosmology example suggests that a ‘supermassive’ orthogonal die underlies the galaxy point patterns of the Universe. The random matrix application reveals that the variance of the spectral gap of the Gaussian orthogonal ensemble exhibits non-monotone scaling with thinning for the Dirac (empirical) random measure, in contrast to the orthogonal dice where the spectral gap variance is monotone. Finally, the approximation application shows that the discrete Legendre polynomials converge to the Charlier polynomials in Lp for integers p ≥ 1 whereas the electronic circuit/shot noise application identifies electric current as the Ornstein-Uhlenbeck process driven by an orthogonal die random *** Codes 60G57, 60G55, 60F05 Copyright © 2020, The Authors. All rights reserved.

关键词： Poisson distribution

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Bayesian Formulations for Graph Spectral Denoising

arXiv

引用

arXiv 2023年

作者： Leone, Sam Sun, Xingzhi Perlmutter, Michael Krishnaswamy, Smita Program for Applied Mathematics Yale University United States Department of Computer Science Yale University United States Department of Mathematics Boise State University United States Department of Genetics Yale School of Medicine United States Wu Tsai Institute Yale University United States FAIR Meta AI United States Computational Biology and Bioinformatics Program Yale University United States

Here we consider the problem of denoising features associated to complex data, modeled as signals on a graph, via a smoothness prior. This is motivated in part by settings such as single-cell RNA where the data is very high-dimensional, but its structure can be captured via an affinity graph. This allows us to utilize ideas from graph signal processing. In particular, we present algorithms for the cases where the signal is perturbed by Gaussian noise, dropout, and uniformly distributed noise. The signals are assumed to follow a prior distribution defined in the frequency domain which favors signals which are smooth across the edges of the graph. By pairing this prior distribution with our three models of noise generation, we propose Maximum A Posteriori (M.A.P.) estimates of the true signal in the presence of noisy data and provide algorithms for computing the M.A.P. Finally, we demonstrate the algorithms' ability to effectively restore signals from white noise on image data and from severe dropout in single-cell RNA sequence data. © 2023, CC BY.

关键词： Frequency domain analysis

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Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets

arXiv

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arXiv 2024年

作者： Landa, Boris Kluger, Yuval Ma, Rong Department of Electrical Engineering Yale University United States Program in Applied Mathematics Yale University United States Department of Biostatistics Harvard University United States Interdepartmental Program in Computational Biology and Bioinformatics Yale University United States Department of Pathology Yale University School of Medicine United States

Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental conditions. Such datasets may share underlying structures of interest but exhibit individual distortions, resulting in misaligned embeddings using traditional techniques. In this work, we propose Entropic Optimal Transport (EOT) eigenmaps, a principled approach for aligning and jointly embedding a pair of datasets with theoretical guarantees. Our approach leverages the leading singular vectors of the EOT plan matrix between two datasets to extract their shared underlying structure and align the datasets accordingly in a common embedding space. We interpret our approach as an inter-data variant of the classical Laplacian eigenmaps and diffusion maps embeddings, showing that it enjoys many favorable analogous properties. We then analyze a data-generative model where two observed high-dimensional datasets share latent variables on a common low-dimensional manifold, but each dataset is subject to data-specific translation, scaling, nuisance structures, and noise. We show that in a high-dimensional asymptotic regime, the EOT plan recovers the shared manifold structure by approximating a kernel function evaluated at the locations of the latent variables. Subsequently, we provide a geometric interpretation of our embedding by relating it to the eigenfunctions of population-level operators encoding the density and geometry of the shared manifold. Finally, we showcase the performance of our approach for data integration and embedding through simulations and analyses of real-world biological data, demonstrating its advantages over alternative methods in challenging scenarios. Copyright © 2024, The Authors. All rights reserved.

关键词： Embeddings

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Velocity-gauge Real-time Time-dependent Density Functional Tight-binding for Large-scale Condensed Matter Systems

arXiv

引用

arXiv 2023年

作者： Xu, Qiang Del Ben, Mauro Okyay, Mahmut Sait Choi, Min Ibrahim, Khaled Z. Wong, Bryan M. Materials Science & Engineering Program Department of Chemistry Department of Physics & Astronomy University of California-Riverside RiversideCA92521 United States Applied Mathematics & Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States

We present a new velocity-gauge real-time, time-dependent density functional tight-binding (VG-rtTDDFTB) implementation in the open-source DFTB+ software package (https://***) for probing electronic excitations in large, condensed matter systems. Our VG-rtTDDFTB approach enables real-time electron dynamics simulations of large, periodic, condensed matter systems containing thousands of atoms with a favorable computational scaling as a function of system size. We provide computational details and benchmark calculations to demonstrate its accuracy and computational parallelizability on a variety of large material systems. As a representative example, we calculate laser-induced electron dynamics in a 512-atom amorphous silicon supercell to highlight the large periodic systems that can be examined with our implementation. Taken together, our VG-rtTDDFTB approach enables new electron dynamics simulations of complex systems that require large periodic supercells, such as crystal defects, complex surfaces, nanowires, and amorphous materials. Copyright © 2023, The Authors. All rights reserved.

关键词： Gages

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