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A lower bound on the complexity of approximate nearest-neighbor searching on the Hamming cube 99

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A lower bound on the complexity of approximate nearest-neigh...

Proceedings of the thirty-first annual ACM symposium on Theory of Computing

作者： Amit Chakrabarti Bernard Chazelle Benjamin Gum Alexey Lvov Department of Computer Science Princeton University Department of Computer Science Princeton University and NEC Research Institute Program in Applied and Computational Mathematics Princeton University

Empowering Optimal Control with Machine Learning: A Perspective from Model Predictive Control

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IFAC-PapersOnLine 2022年第30期55卷 121-126页

作者： E Weinan Jiequn Han Jihao Long AI for Science Institute Beijing Center for Machine Learning Research and School of Mathematical Sciences Peking University Beijing China Center for Computational Mathematics Flatiron Institute New York 10010 USA Program of Applied and Computational Mathematics Princeton University Princeton 08544 USA

Solving complex optimal control problems have confronted computational challenges for a long time. Recent advances in machine learning have provided us with new opportunities to address these challenges. This paper takes model predictive control, a popular optimal control method, as the primary example to survey recent progress that leverages machine learning techniques to empower optimal control solvers. We also discuss some of the main challenges encountered when applying machine learning to develop more robust optimal control algorithms.

关键词： Machine Learning Optimal Control Model Predictive Control

Smooth graph signal interpolation for big data

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arXiv 2018年

作者： Heimowitz, Ayelet Eldar, Yonina C. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Faculty of Math and Computer Science Weizmann Institute of Science Rehovot Israel

In this paper we present the Markov variation, a smoothness measure which offers a probabilistic interpretation of graph signal smoothness. This measure is then used to develop an optimization framework for graph signal interpolation. Our approach is based on diffusion embedding vectors and the connection between diffusion maps and signal processing on graphs. As diffusion embedding vectors may be expensive to compute for large graphs, we present a computationally efficient method, based on the Nyström extension, for interpolation of signals over a graph. We demonstrate our approach on the MNIST dataset and a dataset of daily average temperatures around the US. We show that our method outperforms state of the art graph signal interpolation methods on both datasets, and that our computationally efficient reconstruction achieves slightly reduced accuracy with a large speedup. Copyright © 2018, The Authors. All rights reserved.

关键词： computational efficiency

Adaptive coupling of a deep neural network potential to a classical force field

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arXiv 2018年

作者： Zhang, Linfeng Wang, Han Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Data Science and Beijing International Center for Mathematical Research Peking University China Beijing Institute of Big Data Research Beijing100871 China

An adaptive modeling method (AMM) that couples a deep neural network potential and a classical force field is introduced to address the accuracy-efficiency dilemma faced by the molecular simulation community. The AMM simulated system is decomposed into three types of regions. The first type captures the important phenomena in the system and requires high accuracy, for which we use the Deep Potential Molecular Dynamics (DeePMD) model in this work. The DeePMD model is trained to accurately reproduce the statistical properties of the ab initio molecular dynamics. The second type does not require high accuracy and a classical force field is used to describe it in an efficient way. The third type is used for a smooth transition between the first and the second types of regions. By using a force interpolation scheme and imposing a thermodynamics force in the transition region, we make the DeePMD region embedded in the AMM simulated system as if it were embedded in a system that is fully described by the accurate potential. A representative example of the liquid water system is used to show the feasibility and promise of this method. Copyright © 2018, The Authors. All rights reserved.

关键词： Deep neural networks

MONGE-AMPÈRE FLOW FOR GENERATIVE MODELING

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arXiv 2018年

作者： Zhang, Linfeng Weinan, E. Wang, Lei Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University United States Center for Data Science Peking University Beijing Institute of Big Data Research Beijing100871 China Institute of Physics Chinese Academy of Sciences Beijing100190 China

We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Ampère flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Ampère equation, optimal transport, and fluid dynamics into reversible flow-based generative models. Copyright © 2018, The Authors. All rights reserved.

关键词： Statistical mechanics

Artificial neural network approach for turbulence models: A local framework

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Physical Review Fluids 2021年第8期6卷 084612-084612页

作者： Chenyue Xie Xiangming Xiong Jianchun Wang Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China

A local artificial neural network (LANN) framework is developed for turbulence modeling. The Reynolds-averaged Navier-Stokes (RANS) unclosed terms are reconstructed by the artificial neural network based on the local coordinate system which is orthogonal to the curved walls. We verify the proposed model in the flows over periodic hills. The correlation coefficients of the RANS unclosed terms predicted by the LANN model can be made larger than 0.96 in an a priori analysis, and the relative error of the unclosed terms can be made smaller than 18%. In an a posteriori analysis, detailed comparisons are made on the results of RANS simulations using the LANN, global artificial neural network (GANN), Spalart-Allmaras (SA), and shear stress transport (SST) k−ω models. It is shown that the LANN model performs better than the GANN, SA, and SST k−ω models in the prediction of the average velocity, wall-shear stress, and average pressure, which gives the results that are essentially indistinguishable from the direct numerical simulation data. The LANN model trained at low Reynolds number, Re=2800, can be directly applied to the cases of high Reynolds numbers, Re=5600, 10 595, 19 000, and 37 000, with accurate predictions. Furthermore, the LANN model is verified for flows over periodic hills with varying slopes. These results suggest that the LANN framework has a great potential to be applied to complex turbulent flows with curved walls.

关键词： Compressible boundary layers Turbulence Artificial neural networks

Structure and dynamics of highly charged heavy ions studied with the electron beam ion trap in Tokyo

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Hyperfine Interactions 2011年第1期199卷 123-130页

作者： Nakamura, Nobuyuki Hu, Zhimin Watanabe, Hirofumi Li, Yueming Kato, Daiji Currell, Fred J. Tong, Xiao-Min Watanabe, Tsutomu Ohtani, Shunsuke Institute for Laser Science The University of Electro-Communications 1-5-1 Chofugaoka Chofu Tokyo 182-8585 Japan Chubu University 1200 Matsumoto-cho Kasugai-shi Aichi 487-8501 Japan Institute of Applied Physics and Computational Mathematics P.O.Box 8009 Beijing 100088 China National Institute for Fusion Science 322-6 Oroshi-cho Toki Gifu 509-5292 Japan Centre for Plasma Physics School of Mathematics and Physics Queen's University Belfast Belfast BT7 1NN United Kingdom Doctoral Program in Materials Science Graduate School of Pure and Applied Sciences University of Tsukuba Tsukuba Ibaraki 305-8573 Japan

In this paper, we present the structure and the dynamics of highly charged heavy ions studied through dielectronic recombination (DR) observations performed with the Tokyo electron beam ion trap. By measuring the energy dependence of the ion abundance ratio in the trap at equilibrium, we have observed DR processes for open shell systems very clearly. Remarkable relativistic effects due to the generalized Breit interaction have been clearly shown in DR for highly charged heavy ions. We also present the first result for the coincidence measurement of two photons emitted from a single DR event. © 2011 Springer science+Business Media B.V.

关键词： Heavy ions

NOMAD: nonlinear manifold decoders for operator learning 22

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NOMAD: nonlinear manifold decoders for operator learning

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

作者： Jacob H. Seidman Georgios Kissas Paris Perdikaris George J. Pappas Graduate Program in Applied Mathematics and Computational Science University of Pennsylvania Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Department of Electrical and Systems Engineering University of Pennsylvania

ISBN: (纸本)9781713871088

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation. Here we present NOMAD, a novel operator learning framework with a nonlinear decoder map capable of learning finite dimensional representations of nonlinear submanifolds in function spaces. We show this method is able to accurately learn low dimensional representations of solution manifolds to partial differential equations while outperforming linear models of larger size. Additionally, we compare to state-of-the-art operator learning methods on a complex fluid dynamics benchmark and achieve competitive performance with a significantly smaller model size and training cost.

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Active learning of uniformly accurate inter-atomic potentials for materials simulation

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arXiv 2018年

作者： Zhang, Linfeng Lin, De-Ye Wang, Han Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China CAEP Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China 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 Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

An active learning procedure called Deep Potential Generator (DP-GEN) is proposed for the construction of accurate and transferable machine learning-based models of the potential energy surface (PES) for the molecular modeling of materials. This procedure consists of three main components: exploration, generation of accurate reference data, and training. Application to the sample systems of Al, Mg and Al-Mg alloys demonstrates that DP-GEN can produce uniformly accurate PES models with a minimal number of reference data. Copyright © 2018, The Authors. All rights reserved.

关键词： Potential energy