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Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations

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arXiv 2024年

作者： Lee, Jae Yong Schotthöfer, Steffen Xiao, Tianbai Krumscheid, Sebastian Frank, Martin Center for Artificial Intelligence and Natural Sciences Korea Institute for Advanced Study Seoul Korea Republic of Oak Ridge National Laboratory Computer Science and Mathematics Division Oak RidgeTN United States State Key Laboratory of High Temperature Gas Dynamics Centre for Interdisciplinary Research in Fluids Institute of Mechanics Chinese Academy of Sciences Beijing China School of Engineering Science University of Chinese Academy of Sciences Beijing China Karlsruhe Institute of Technology Scientific Computing Center Karlsruhe76131 Germany

This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision operator, which is a powerful yet expensive mathematical model for interacting particle systems with ample applications, e.g., in radiation transport. We investigate the capabilities of the Deep Operator network (DeepONet) approach to modelling the high dimensional collision operator of the linear kinetic equation. This integral operator has crucial analytical structures that a surrogate model, e.g., a DeepONet, needs to preserve to enable meaningful physical simulation. We propose several DeepONet modifications to encapsulate essential structural properties of this integral operator in a DeepONet model. To be precise, we adapt the architecture of the trunk-net so the DeepONet has the same collision invariants as the theoretical kinetic collision operator, thus preserving conserved quantities, e.g., mass, of the modeled many-particle system. Further, we propose an entropy-inspired data-sampling method tailored to train the modified DeepONet surrogates without requiring an excessive expensive simulation-based data generation. © 2024, CC BY-NC-SA.

关键词： Kinetic energy

Non-invasive assessment of intracranial wall shear stress using high-resolution magnetic resonance imaging in combination with computational fluid dynamics technique

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Fundamental Research 2022年第2期2卷 329-334页

作者： Yuwen Chen Jia Liu Mingli Li Yannan Yu Zhengzheng Yan Wenshin Shiu Bokai Wu Zaiheng Cheng Yao Meng Yang Liu Weizhuang Yuan Zongmuyu Zhang Weihai Xu Department of Neurology State Key Laboratory of Complex Severe and Rare DiseasesPeking Union Medical Cllege HospitalChinese Academy of Medical Sciences and Peking Union Medical College1 ShuaifuyuanBeijing 100005China Department of Cardiology Second Afliated Hospital of Zhejiang University School of MedicineHangzhouZhejiang 310009China Shenzhen Instiutes of Advanced Technology Chinese Academy of SciencesShenzhen 518055China Shenzhen Key Laboratory for Exascale Engineering and Scientific Computing Shenzhen 518055China Department of Radiology Peking Union Medical College HospitalChinese Acadermy of Medical Sciences1 ShuaifiryuanBeijing 100005China Department of Radiology Stanford UniversityCaliformia 94305United States Department of Neurology Saarland UniversityKirrberger StraSe 66421Germany

In vivo studies on association between wall shear stress(WSS)and intracranial plaque are *** on the three-dimensional T1-weighted high-resolution magnetic resonance imaging(3DT1 HR-MRI)data of patients with low-grade stenotic(<50%)atherosclerotic middle cerebral artery(MCA)and subjects with normal MCA,we built a three-dimensional reconstructed WSS model by computational fluid dynamics(CFD)***-dimensional registration of the CFD model to the HR-MRI was performed with projections based on the resolution and thickness of the *** relationships between the Wss at each side of the vessel wall and plaque location were analyzed.A total of 94 MCA plaques from 43 patients and 50 normal MCAs were *** the normal MCAs,WSS was lower at the ventral-inferior wall than at the dorsal-superior wall(proximal segment,p<0.001;middle segment,p<0.001)and lower at the inner wall than at the outer wall of the MCA curve(p<0.001).In atherosclerotic MCAs,similar low Wss regions were observed where plaques *** WSS ratio of the ventral-inferior wall to the dorsal-superior wall in atherosclerotic MCAs was lower than that in normal MCAs(p=0.002).The WSS_(inmer-outer)ratio in atherosclerotic MCAs was lower than that in normal MCAs(p=0.002).Low WSS was associated with MCA atherosclerosis formation and occurred mainly at the ventral inferior wall,which was anatomically opposite the orifices of penetrating arteries,and at the inner wall of the MCA ***,the results were well consistent with the low WSS theory in atherosclerosis *** reconstructed WSS model is a promising novel method for assessing an individualized vascular profile once validated by further studies.

关键词： Intracranial atherosclerosis Wall shear stress Atherosclerotic plaque High-resolution magnetic resonance imaging Computational fluid dynamics

Unifying Farkas lemma and S-lemma: New theory and applications in nonquadratic nonconvex optimization

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arXiv 2021年

作者： Yang, Meijia Xia, Yong Wang, Shu School of Mathematics and Physics University of Science and Technology Beijing Beijing100083 China State Key Laboratory of Software Development Environment LMIB of the Ministry of Education School of Mathematics and System Sciences Beihang University Beijing100191 China Institute of Computational Mathematics and Scientific/Engineering Computing State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We unify nonlinear Farkas lemma and S-lemma to a generalized alternative theorem for nonlinear nonconvex system. It provides fruitful applications in globally solving nonconvex non-quadratic optimization problems via ... 详细信息

关键词： Quadratic programming

Adam-family Methods for Nonsmooth Optimization with Convergence Guarantees

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arXiv 2023年

作者： Xiao, Nachuan Hu, Xiaoyin Liu, Xin Toh, Kim-Chuan Institute of Operations Research and Analytics National University of Singapore 3 Research Link Singapore117602 Singapore School of Computer and Computing Science Hangzhou City University Hangzhou310015 China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Mathematics Institute of Operations Research and Analytics National University of Singapore 10 Lower Kent Ridge Road Singapore119076 Singapore

In this paper, we present a comprehensive study on the convergence properties of Adam-family methods for nonsmooth optimization, especially in the training of nonsmooth neural networks. We introduce a novel two-timescale framework that adopts a two-timescale updating scheme, and prove its convergence properties under mild assumptions. Our proposed framework encompasses various popular Adam-family methods, providing convergence guarantees for these methods in training nonsmooth neural networks. Furthermore, we develop stochastic subgradient methods that incorporate gradient clipping techniques for training nonsmooth neural networks with heavy-tailed noise. Through our framework, we show that our proposed methods converge even when the evaluation noises are only assumed to be integrable. Extensive numerical experiments demonstrate the high efficiency and robustness of our proposed methods. Copyright © 2023, The Authors. All rights reserved.

关键词： Stochastic systems

Tightness and equivalence of semidefinite relaxations for mimo detection

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arXiv 2021年

作者： Jiang, Ruichen Liu, Ya-Feng Bao, Chenglong Jiang, Bo Department of Electronic Engineering Tsinghua University Beijing100084 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Yau Mathematical Sciences Center Tsinghua University Beijing100084 China School of Mathematical Sciences Key Laboratory for NSLSCS of Jiangsu Province Nanjing Normal University Nanjing210023 China

The multiple-input multiple-output (MIMO) detection problem, a fundamental problem in modern digital communications, is to detect a vector of transmitted symbols from the noisy outputs of a fading MIMO channel. The maximum likelihood detector can be formulated as a complex least-squares problem with discrete variables, which is NP-hard in general. Various semidefinite relaxation (SDR) methods have been proposed in the literature to solve the problem due to their polynomial-time worst-case complexity and good detection error rate performance. In this paper, we consider two popular classes of SDR-based detectors and study the conditions under which the SDRs are tight and the relationship between different SDR models. For the enhanced complex and real SDRs proposed recently by Lu et al., we refine their analysis and derive the necessary and sufficient condition for the complex SDR to be tight, as well as a necessary condition for the real SDR to be tight. In contrast, we also show that another SDR proposed by Mobasher et al. is not tight with high probability under mild conditions. Moreover, we establish a general theorem that shows the equivalence between two subsets of positive semidefinite matrices in different dimensions by exploiting a special "separable" structure in the constraints. Our theorem recovers two existing equivalence results of SDRs defined in different settings and has the potential to find other applications due to its *** Codes 90C22, 90C20, 90C46, 90C27 Copyright © 2021, The Authors. All rights reserved.

关键词： Polynomial approximation

Splitting Method for Stochastic Navier-Stokes Equations

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arXiv 2025年

作者： Zhu, Jie Zhu, Yujun Ming, Ju Gunzburger, Max D. School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Department of Scientific Computing Florida State University TallahasseeFL32304 United States

This paper investigates the two-dimensional stochastic steady-state Navier-Stokes(NS) equations with additive random noise. We introduce an innovative splitting method that decomposes the stochastic NS equations into a deterministic NS component and a stochastic equation. We rigorously analyze the proposed splitting method from the perspectives of equivalence, stability, existence and uniqueness of the solution. We also propose a modified splitting scheme, which simplified the stochastic equation by omitting its nonlinear terms. A detailed analysis of the solution properties for this modified approach is provided. Additionally, we discuss the statistical errors with both the original splitting format and the modified scheme. Our theoretical and numerical studies demonstrate that the equivalent splitting scheme exhibits significantly enhanced stability compared to the original stochastic NS equations, enabling more effective handling of nonlinear characteristics. Several numerical experiments were performed to compare the statistical errors of the splitting method and the modified splitting method. Notably, the deterministic NS equation in the splitting method does not require repeated solving, and the stochastic equation in the modified scheme is free of nonlinear terms. These features make the modified splitting method particularly advantageous for large-scale computations, as it significantly improves computational efficiency without compromising accuracy. Copyright © 2025, The Authors. All rights reserved.

关键词： Navier Stokes equations

Dissolving Constraints for Riemannian Optimization

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arXiv 2022年

作者： Xiao, Nachuan Liu, Xin Toh, Kim-Chuan The Institute of Operations Research and Analytics National University of Singapore Singapore State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China Department of Mathematics Institute of Operations Research and Analytics National University of Singapore Singapore119076 Singapore

In this paper, we consider optimization problems over closed embedded submanifolds of Rn, which are defined by the constraints c(x) = 0. We propose a class of constraint dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint dissolving function named CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and Lojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint dissolving approaches. Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization

A global optimization approach for multi-marginal optimal transport problems with Coulomb cost

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arXiv 2021年

作者： Hu, Yukuan Chen, Huajie Liu, Xin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China School of Mathematical Sciences Beijing Normal University China

In this work, we construct a novel numerical method for solving the multi-marginal optimal transport problems with Coulomb cost. This type of optimal transport problems arises in quantum physics and plays an important role in understanding the strongly correlated quantum systems. With a Monge-like ansatz, the orginal high-dimensional problems are transferred into mathematical programmings with generalized complementarity constraints, and thus the curse of dimensionality is surmounted. However, the latter ones are themselves hard to deal with from both theoretical and practical perspective. Moreover in the presence of nonconvexity, brute-force searching for global solutions becomes prohibitive as the problem size grows large. To this end, we propose a global optimization approach for solving the nonconvex optimization problems, by exploiting an efficient proximal block coordinate descent local solver and an initialization subroutine based on hierarchical grid refinements. We provide numerical simulations on some typical physical systems to show the efficiency of our approach. The results match well with both theoretical predictions and physical intuitions, and give the first visualization of optimal transport maps for some two dimensional *** Codes 81V05 (Primary) 49M37, 65K05, 90C26, 90C30 (Secondary) Copyright © 2021, The Authors. All rights reserved.

关键词： Global optimization

A force-based gradient descent method for ab initio atomic structure relaxation

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arXiv 2022年

作者： Hu, Yukuan Gao, Xingyu Zhao, Yafan Liu, Xin Song, Haifeng State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China

Force-based algorithms for ab initio atomic structure relaxation, such as conjugate gradient methods, usually get stuck in the line minimization processes along search directions, where expensive ab initio calculations are triggered frequently to test trial positions before locating the next iterate. We present a force-based gradient descent method, WANBB, that circumvents the deficiency. At each iteration, WANBB enters the line minimization process with a trial stepsize capturing the local curvature of the energy surface. The exit is controlled by an unrestrictive criterion that tends to accept early trials. These two ingredients streamline the line minimization process in WANBB. The numerical simulations on nearly 80 systems with good universality demonstrate the considerable compression of WANBB on the cost for the unaccepted trials compared with conjugate gradient methods. We also observe across the board significant and universal speedups as well as the superior robustness of WANBB over several widely used methods. The latter point is theoretically established. The implementation of WANBB is pretty simple, in that no a priori physical knowledge is required and only two parameters are present without tuning. © 2022, CC BY.

关键词： Conjugate gradient method