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Machine learning based non-Newtonian fluid model with molecular fidelity

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arXiv 2020年

作者： Lei, Huan Wu, Lei Weinan, E. Department of Computational Mathematics Science & Engineering Department of Statistics & Probability Michigan State University MI48824 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We introduce a machine-learning-based framework for constructing continuum non-Newtonian fluid dynamics model directly from a micro-scale description. Polymer solution is used as an example to demonstrate the essential ideas. To faithfully retain molecular fidelity, we establish a micro-macro correspondence via a set of encoders for the micro-scale polymer configurations and their macro-scale counterparts, a set of nonlinear conformation tensors. The dynamics of these conformation tensors can be derived from the micro-scale model and the relevant terms can be parametrized using machine learning. The final model, named the deep non-Newtonian model (DeePN2), takes the form of conventional non-Newtonian fluid dynamics models, with a new form of the objective tensor derivative. Numerical results demonstrate the accuracy of DeePN2. Copyright © 2020, The Authors. All rights reserved.

关键词： Viscous flow

Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization

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Computation of sparse low degree interpolating polynomials a...

作者： Bandeira, A.S. Scheinberg, K. Vicente, L.N. Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Department of Industrial and Systems Engineering Lehigh University Harold S. Mohler Laboratory 200 West Packer Avenue Bethlehem PA 18015-1582 United States CMUC Department of Mathematics University of Coimbra 3001-501 Coimbra Portugal

Interpolation-based trust-region methods are an important class of algorithms forDerivative-Free Optimizationwhich rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations between variables are negligible, implying, in the smooth case, a sparse structure in the Hessian matrix. To be able to exploit Hessian sparsity, existing optimization approaches require the knowledge of the sparsity structure. The goal of this paper is to develop and analyze a method where the sparse models are constructed automatically. The sparse recovery theory developed recently in the field of compressed sensing characterizes conditions under which a sparse vector can be accurately recovered from few random measurements. Such a recovery is achieved by minimizing the l1-norm of a vector subject to the measurements constraints. We suggest an approach for building sparse quadratic polynomial interpolationmodels byminimizing the l1-norm of the entries of themodel Hessian subject to the interpolation conditions. We show that this procedure recovers accurate models when the function Hessian is sparse, using relatively few randomly selected sample points. Motivated by this result, we developed a practical interpolation-based trust-region method using deterministic sample sets and minimum l1-norm quadratic models. Our computational results show that the new approach exhibits a promising numerical performance both in the general case and in the sparse one. © Springer and Mathematical Optimization Society 2012.

关键词： Interpolation

A SINGULAR TWO-PHASE STEFAN PROBLEM AND PARTICLES INTERACTING THROUGH THEIR HITTING TIMES

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arXiv 2022年

作者： Baker, Graeme Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance and Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension that can be found in [DIRT15a], [NS19a], [HLS18], [CRS20]. As therein, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not monotone, so that the large system limit is obtained by a novel *** Codes 82C22, 60H30, 35K10, 35K60, 91G80 Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

Nonbacktracking bounds on the influence in independent cascade models

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arXiv 2017年

作者： Abbe, Emmanuel Kulkarni, Sanjeev Lee, Eun Jee Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University Princeton United States Department of Electrical Engineering Princeton University Princeton United States Program in Applied and Computational Mathematics Princeton University Princeton United States

This paper develops upper and lower bounds on the influence measure in a network, more precisely, the expected number of nodes that a seed set can influence in the independent cascade model. In particular, our bounds exploit nonbacktracking walks, Fortuin–Kasteleyn–Ginibre (FKG) type inequalities, and are computed by message passing implementation. Nonbacktracking walks have recently allowed for headways in community detection, and this paper shows that their use can also impact the influence computation. Further, we provide a knob to control the trade-off between the efficiency and the accuracy of the bounds. Finally, the tightness of the bounds is illustrated with simulations on various network models. Copyright © 2017, The Authors. All rights reserved.

关键词： Economic and social effects

WELL-POSEDNESS OF THE SUPERCOOLED STEFAN PROBLEM WITH OSCILLATORY INITIAL CONDITIONS

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arXiv 2023年

作者： Mustapha, Scander Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance and Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We study the one-phase one-dimensional supercooled Stefan problem with oscillatory initial conditions. In this context, the global existence of so-called physical solutions has been shown recently in [CRSF23], despite the presence of blow-ups in the freezing rate. On the other hand, for regular initial conditions, the uniqueness of physical solutions has been established in [DNS22]. Here, we prove the uniqueness of physical solutions for oscillatory initial conditions by a new contraction argument that replaces the local monotonicity condition of [DNS22] by an averaging condition. We verify this weaker condition for fairly general oscillating probability densities, such as the ones given by an almost sure trajectory of (1 + Wx - √2x| log | log x||)+ ^ 1 near the origin, where W is a standard Brownian motion. We also permit typical deterministically constructed oscillating densities, including those of the form (1 + sin 1/x)/2 near the origin. Finally, we provide an example of oscillating densities for which it is possible to go beyond our main assumption via further complementary *** Codes 80A22, 35B05, 60H30 Copyright © 2023, The Authors. All rights reserved.

关键词： Brownian movement

Sigma-delta quantization and finite frames

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Sigma-delta quantization and finite frames

International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： J.J. Benedetto O. Yilmaz A.M. Powell Department of Mathematics University of Maryland College Park MD USA Program in Applied and Computational Mathematics Princeton University Princeton NJ USA

It is shown that sigma-delta (/spl Sigma//spl Delta/) algorithms can be used effectively to quantize finite frame expansions for R/sup d/. Error estimates for various quantized frame expansions are derived, and, in pa... 详细信息

关键词： Delta-sigma modulation Quantization Hilbert space mathematics Signal processing algorithms Phase change materials Signal processing Dictionaries Educational institutions Digital signal processing

Isotope effects in x-ray absorption spectra of liquid water

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Physical Review B 2020年第11期102卷 115155-115155页

作者： Chunyi Zhang Linfeng Zhang Jianhang Xu Fujie Tang Biswajit Santra Xifan Wu Department of Physics Temple University Philadelphia Pennsylvania 19122 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The isotope effects in x-ray absorption spectra of liquid water are studied by a many-body approach within electron-hole excitation theory. The molecular structures of both light and heavy water are modeled by path-integral molecular dynamics based on the advanced deep-learning technique. The neural network is trained on ab initio data obtained with SCAN density functional theory. The experimentally observed isotope effect in x-ray absorption spectra is reproduced semiquantitatively in theory. Compared to the spectrum in normal water, the blueshifted and less pronounced pre- and main-edge in heavy water reflect that the heavy water is more structured at short- and intermediate-range of the hydrogen-bond network. In contrast, the isotope effect on the spectrum is negligible at post-edge, which is consistent with the identical long-range ordering in both liquids as observed in the diffraction experiment.

关键词： Atomic & molecular structure Electronic excitation & ionization Water Density functional calculations Electronic structure First-principles calculations Machine learning Molecular dynamics X-ray absorption spectroscopy

DeePKS-kit: A package for developing machine learning-based chemically accurate energy and density functional models

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arXiv 2020年

作者： Chen, Yixiao Zhang, Linfeng Wang, Han Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Mathematics Princeton University PrincetonNJ United States

We introduce DeePKS-kit, an open-source software package for developing machine learning based energy and density functional models. DeePKS-kit is interfaced with PyTorch, an open-source machine learning library, and PySCF, an ab initio computational chemistry program that provides simple and customized tools for developing quantum chemistry codes. It supports the DeePHF and DeePKS methods. In addition to explaining the details in the methodology and the software, we also provide an example of developing a chemically accurate model for water clusters. Copyright © 2020, The Authors. All rights reserved.

关键词： Electronic structure

Ground state energy functional with Hartree-Fock efficiency and chemical accuracy

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arXiv 2020年

We introduce the Deep Post–Hartree–Fock (DeePHF) method, a machine learning-based scheme for constructing accurate and transferable models for the ground-state energy of electronic structure problems. DeePHF predicts the energy difference between results of highly accurate models such as the coupled cluster method and low accuracy models such as the the Hartree-Fock (HF) method, using the ground-state electronic orbitals as the input. It preserves all the symmetries of the original high accuracy model. The added computational cost is less than that of the reference HF or DFT and scales linearly with respect to system size. We examine the performance of DeePHF on organic molecular systems using publicly available datasets and obtain the state-of-art performance, particularly on large datasets. Copyright © 2020, The Authors. All rights reserved.

关键词： Molecular physics