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arXiv 2021年

作者： Zhang, Linfeng Wang, Han Muniz, Maria Carolina Panagiotopoulos, Athanassios Z. Car, Roberto Weinan, E. DP Technology Beijing China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China HEDPS CAPT College of Engineering Peking University Beijing100871 China Department of Chemical and Biological Engineering Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States School of Mathematical Sciences Peking University Beijing100871 China AI for Science Institute Beijing China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Machine learning models for the potential energy of multi-atomic systems, such as the deep potential (DP) model, make possible molecular simulations with the accuracy of quantum mechanical density functional theory, at a cost only moderately higher than that of empirical force fields. However, the majority of these models lack explicit long-range interactions and fail to describe properties that derive from the Coulombic tail of the forces. To overcome this limitation we extend the DP model by approximating the long-range electrostatic interaction between ions (nuclei+core electrons) and valence electrons with that of distributions of spherical Gaussian charges located at ionic and electronic sites. The latter are rigorously defined in terms of the centers of the maximally localized Wannier distributions, whose dependence on the local atomic environment is modeled accurately by a deep neural network. In the deep potential long-range (DPLR) model, the electrostatic energy of the Gaussian charge system is added to short-range interactions that are represented as in the standard DP model. The resulting potential energy surface is smooth and possesses analytical forces and virial. Missing effects in the standard DP scheme are recovered, improving on accuracy and predictive power. By including long-range electrostatics, DPLR correctly extrapolates to large systems the potential energy surface learned from quantum mechanical calculations on smaller systems. We illustrate the approach with three examples, the potential energy profile of the water dimer, the free energy of interaction of a water molecule with a liquid water slab, and the phonon dispersion curves of the NaCl crystal. © 2021, CC BY.

关键词： Electrostatics

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Multilevel Particle Filters for Partially Observed McKean-Vlasov Stochastic Differential Equations

arXiv

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arXiv 2024年

作者： Awadelkarim, Elsiddig Jasra, Ajay Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia School of Data Science The Chinese University of Hong Kong Shenzhen China

In this paper we consider the filtering problem associated to partially observed McKean-Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times and the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem, even the ordinary SDE case is challenging and requires numerical approximations. Based upon the ideas in [3, 12] we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the afore-mentioned expectations. We prove under assumptions that, for ϵ > 0, to obtain a mean square error of O(ϵ2) the PF has a cost per-observation time of O(ϵ−5) and the MLPF costs O(ϵ−4) (best case) or O(ϵ−4 log(ϵ)2) (worst case). Our theoretical results are supported by numerical experiments. © 2024, CC BY.

关键词： Mean square error

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Deep Potentials for Materials science

arXiv

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arXiv 2022年

作者： Wen, Tongqi Zhang, Linfeng Wang, Han Weinan, E. Srolovitz, David J. Department of Mechanical Engineering The University of Hong Kong Hong Kong DP Technology Beijing China AI for Science Institute Beijing China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing China HEDPS CAPT College of Engineering Peking University Beijing China School of Mathematical Sciences Peking University Beijing China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton United States Shenzhen China

To fill the gap between accurate (and expensive) ab initio calculations and efficient atomistic simulations based on empirical interatomic potentials, a new class of descriptions of atomic interactions has emerged and been widely applied;i.e., machine learning potentials (MLPs). One recently developed type of MLP is the Deep Potential (DP) method. In this review, we provide an introduction to DP methods in computational materials science. The theory underlying the DP method is presented along with a step-by-step introduction to their development and use. We also review materials applications of DPs in a wide range of materials systems. The DP Library provides a platform for the development of DPs and a database of extant DPs. We discuss the accuracy and efficiency of DPs compared with ab initio methods and empirical potentials. Copyright © 2022, The Authors. All rights reserved.

关键词： Calculations

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Improving Bayesian local spatial models in large data sets

arXiv

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arXiv 2019年

作者： Lenzi, Amanda Castruccio, Stefano Rue, Håvard Genton, Marc G. Statistics Program King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

Environmental processes resolved at a sufficiently small scale in space and time will inevitably display non-stationary behavior. Such processes are both challenging to model and computationally expensive when the data size is large. Instead of modeling the global non-stationarity explicitly, local models can be applied to disjoint regions of the domain. The choice of the size of these regions is dictated by a bias-variance trade-off;large regions will have smaller variance and larger bias, whereas small regions will have higher variance and smaller bias. From both the modeling and computational point of view, small regions are preferable to better accommodate the non-stationarity. However, in practice, large regions are necessary to control the variance. We propose a novel Bayesian three-step approach that allows for smaller regions without compromising the increase of the variance that would follow. We are able to propagate the uncertainty from one step to the next without issues caused by reusing the data. The improvement in inference also results in improved prediction, as our simulated example shows. We illustrate this new approach on a data set of simulated high-resolution wind speed data over Saudi Arabia. Copyright © 2019, The Authors. All rights reserved.

关键词： Wind

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Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior

Gaussian Process Port-Hamiltonian Systems: Bayesian Learning...

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IEEE Conference on Decision and Control

作者： Thomas Beckers Jacob Seidman Paris Perdikaris George J. Pappas Department of Electrical and Systems Engineering University of Pennsylvania Philadelphia PA USA graduate program in Applied Mathematics and Computational Science University of Pennsylvania Philadelphia PA USA Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia PA USA

ISBN: (数字)9781665467612

ISBN: (纸本)9781665467629

Data-driven approaches achieve remarkable results for the modeling of complex dynamics based on collected data. However, these models often neglect basic physical principles which determine the behavior of any real-world system. This omission is unfavorable in two ways: The models are not as data-efficient as they could be by incorporating physical prior knowledge, and the model itself might not be physically correct. We propose Gaussian Process Port-Hamiltonian systems (GPPHS) as a physics-informed Bayesian learning approach with uncertainty quantification. The Bayesian nature of GP-PHS uses collected data to form a distribution over all possible Hamiltonians instead of a single point estimate. Due to the underlying physics model, a GP-PHS generates passive systems with respect to designated inputs and outputs. Further, the proposed approach preserves the compositional nature of Port-Hamiltonian systems.

关键词： Uncertainty Gaussian processes Probabilistic logic Data models Bayes methods Behavioral sciences Kernel

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Classical many-particle systems with unique disordered ground states

arXiv

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arXiv 2017年

作者： Zhang, G. Stillinger, F.H. Torquato, S. Department of Chemistry Princeton University PrincetonNJ08544 Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544

Classical ground states (global energy-minimizing configurations) of many-particle systems arefitypically unique crystalline structures, implying zero enumeration entropy of distinct patterns (aside from trivial symmetry operations). By contrast, the few previously known disordered classical ground states of many-particle systems are all high-entropy (highly degenerate) states. Here we show computationally that our recently-proposed \perfect-glass" many-particle model [Sci. Rep., 6, 36963 (2016)] possesses disordered classical ground states with a zero entropy: a highly counterintuitive situation. For all of the system sizes, parameters, and space dimensions that we have numerically investigated, the disordered ground states are unique such that they can always be superposed onto each other or their mirror image. At low energies, the density of states obtained from simulations matches those calculated from the harmonic approximation near a single ground state, further confirming ground-state uniqueness. Our discovery provides singular examples in which entropy and disorder are at odds with one another. The zero-entropy ground states provide a unique perspective on the celebrated Kauzmann-entropy crisis in which the extrapolated entropy of a supercooled liquid drops below that of the crystal. We expect that our disordered unique patterns to be of value infields beyond glass physics, including applications in cryptography as pseudo-random functions with tunable computational complexity. Copyright © 2017, The Authors. All rights reserved.

关键词： Ground state

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The structure factor of primes

arXiv

引用

arXiv 2018年

作者： Zhang, G. Martelli, F. Torquato, S. Department of Chemistry Princeton University PrincetonNJ08544 Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544

Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical-mechanical methods, especially the structure factor S(k) in an interval M ≤ p ≤ M + L with M large, and L/M smaller than unity. We show that the structure factor of the prime-number configurations in such intervals exhibits well-defined Bragg-like peaks along with a small "diffuse" contribution. This indicates that the primes are appreciably more correlated and ordered than previously thought. Our numerical results definitively suggest an explicit formula for the locations and heights of the peaks. This formula predicts infinitely many peaks in any non-zero interval, similar to the behavior of quasicrystals. However, primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational. We also show numerically that the diffuse part decays slowly as M and L increases. This suggests that the diffuse part vanishes in an appropriate infinite-system-size limit. Copyright © 2018, The Authors. All rights reserved.

关键词： Quasicrystals

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Forecasting high-frequency spatio-temporal wind power with dimensionally reduced echo state networks

arXiv

引用

arXiv 2021年

作者： Huang, Huang Castruccio, Stefano Genton, Marc G. Statistics Program King Abdullah University of Science and Technology Thuwal 23955-6900 Saudi Arabia Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

Fast and accurate hourly forecasts of wind speed and power are crucial in quan-tifying and planning the energy budget in the electric grid. Modeling wind at a high resolution brings forth considerable challenges given its turbulent and highly nonlin-ear dynamics. In developing countries, where wind farms over a large domain are currently under construction or consideration, this is even more challenging given the necessity of modeling wind over space as well. In this work, we propose a machine learning approach to model the nonlinear hourly wind dynamics in Saudi Arabia with a domain-specific choice of knots to reduce the spatial dimensionality. Our re-sults show that for locations highlighted as wind abundant by a previous work, our approach results in an 11% improvement in the two-hour-ahead forecasted power against operational standards in the wind energy sector, yielding a saving of nearly one million US dollars over a year under current market prices in Saudi Arabia. © 2021, CC BY.

关键词： Wind power

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Multi-index Sequential Monte Carlo ratio estimators for Bayesian Inverse problems

arXiv

引用

arXiv 2022年

作者： Law, Kody J.H. Walton, Neil Yang, Shangda Jasra, Ajay School of Mathematics University of Manchester ManchesterM13 9PL United Kingdom Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of Sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of MSE−1, while single level methods require MSE−ξ for ξ > 1. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where ξ = 5/4 and ξ = 3/2, respectively. It is also illustrated on more challenging log Gaussian process models, where single level complexity is approximately ξ = 9/4 and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives ξ = 5/4 + ω, for any ω > 0, whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the spectral domain, which facilitates acceleration with fast Fourier transform methods via a cumulant embedding strategy, and may be of independent interest in the context of spatial statistics and machine learning. Copyright © 2022, The Authors. All rights reserved.

关键词： Inverse problems

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End-To-End Quantum Machine Learning Implemented with Controlled Quantum Dynamics

引用

Physical Review applied 2020年第6期14卷 064020-064020页

作者： Re-Bing Wu Xi Cao Pinchen Xie Yu-xi Liu Department of Automation Tsinghua University Beijing 100084 China Beijing National Research Center for Information Science and Technology Beijing 100084 China Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Institute of Microelectronics Tsinghua University Beijing 100084 China

To achieve quantum machine learning on imperfect noisy intermediate-scale quantum (NISQ) processors, the entire physical implementation should include as few as possible hand-designed modules with only a few ad hoc parameters to be determined. This work presents such a hardware-friendly end-to-end quantum-machine-learning scheme that can be implemented with imperfect NISQ processors. The proposal transforms the machine-learning task to the optimization of controlled quantum dynamics, in which the learning model is parameterized by experimentally tunable control variables. Our design also enables automated feature selection by encoding the raw input to quantum states through agent control variables. Comparing with the gate-based parameterized quantum circuits, the proposed end-to-end quantum-learning model is easy to implement as there are only a few ad hoc parameters to be determined. Numerical simulations on the benchmarking MNIST (Mixed National Institute of Standards and Technology) dataset of handwritten digits demonstrate that the model can achieve high performance using only 3–5 qubits without downsizing the dataset, which shows great potential for accomplishing large-scale real-world learning tasks on NISQ processors.

关键词： Machine learning Noise Quantum control Quantum information architectures & platforms Artificial neural networks

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