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Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems

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Communications in Computational Physics 2023年第2期33卷 596-627页

作者： Sidi Wu Aiqing Zhu Yifa Tang Benzhuo Lu State Key Laboratory of Scientific and Engineering Computing National Center for Mathematics and Interdisciplinary SciencesAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China.

With the remarkable empirical success of neural networks across diverse scientific disciplines,rigorous error and convergence analysis are also being developed and ***,there has been little theoretical work focusing on neural networks in solving interface *** this paper,we perform a convergence analysis of physics-informed neural networks(PINNs)for solving second-order elliptic interface ***,we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump *** is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in H2 as the number of samples *** experiments are provided to demonstrate our theoretical analysis.

关键词： Elliptic interface problems generalization errors convergence analysis neural networks.

A NEW ADAPTIVE SUBSPACE MINIMIZATION THREE-TERM CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION

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Journal of Computational Mathematics 2021年第2期39卷 159-177页

作者： Keke Zhang Hongwei Liu Zexian Liu School of Mathematics and Statistics Xidian UniversityXi’an 710126China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering computingAMSSChinese Academy of SciencesBeijing 100190China

A new adaptive subspace minimization three-term conjugate gradient algorithm with nonmonotone line search is introduced and analyzed in this *** search directions are computed by minimizing a quadratic approximation of the objective function on special subspaces,and we also proposed an adaptive rule for choosing different searching directions at each *** obtain a significant conclusion that the each choice of the search directions satisfies the sufficient descent *** the used nonmonotone line search,we prove that the new algorithm is globally convergent for general nonlinear functions under some mild *** experiments show that the proposed algorithm is promising for the given test problem set.

关键词： Conjugate gradient method Nonmonotone line search Subspace minimization Sufficient descent condition Global convergence

A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints

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arXiv 2025年

作者： Yang, Yan Gao, Bin Yuan, Ya-Xiang State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences The University of Chinese Academy of Sciences Beijing China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Imposing additional constraints on low-rank optimization has garnered growing interest. However, the geometry of coupled constraints hampers the well-developed low-rank structure and makes the problem intricate. To this end, we propose a space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints. The "space-decoupling" is reflected in several ways. We show that the tangent cone of coupled constraints is the intersection of tangent cones of each constraint. Moreover, we decouple the intertwined bounded-rank and orthogonally invariant constraints into two spaces, leading to optimization on a smooth manifold. Implementing Riemannian algorithms on this manifold is painless as long as the geometry of additional constraints is known. In addition, we unveil the equivalence between the reformulated problem and the original problem. Numerical experiments on real-world applications—spherical data fitting, graph similarity measuring, low-rank SDP, model reduction of Markov processes, reinforcement learning, and deep learning—validate the superiority of the proposed framework. Copyright © 2025, The Authors. All rights reserved.

关键词： Markov processes

Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

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Communications on Applied Mathematics and Computation 2024年第1期6卷 30-63页

作者： Maohui Lyu Vrushali A.Bokil Yingda Cheng Fengyan Li State Key Laboratory of Scientific and Engineering Computing(LSEC) Academy of Mathematics and System SciencesChinese Academy of SciencesBeijing100190China Department of Mathematics College of ScienceOregon State UniversityCorvallisOR97331USA Department of Mathematics Michigan State UniversityEast LansingMI48824USA Department of Computational Mathematics Science and EngineeringMichigan State UniversityEast LansingMI48824USA Department of Mathematical Sciences Rensselaer Polytechnic InstituteTroyNY12180USA

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational *** the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the *** main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order *** with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel *** evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

关键词： Maxwell’s equations Kerr and Raman Discontinuous Galerkin method Energy stability

Desingularization of bounded-rank tensor sets

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arXiv 2024年

作者： Gao, Bin Peng, Renfeng Yuan, Ya-Xiang State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences Beijing China

Low-rank tensors appear to be prosperous in many applications. However, the sets of bounded-rank tensors are non-smooth and non-convex algebraic varieties, rendering the low-rank optimization problems to be challenging. To this end, we delve into the geometry of bounded-rank tensor sets, including Tucker and tensor train formats. We propose a desingularization approach for bounded-rank tensor sets by introducing slack variables, resulting in a low-dimensional smooth manifold embedded in a higher-dimensional space while preserving the structure of low-rank tensor formats. Subsequently, optimization on tensor varieties can be reformulated to optimization on smooth manifolds, where the methods and convergence are well explored. We reveal the relationship between the landscape of optimization on varieties and that of optimization on manifolds. Numerical experiments on tensor completion illustrate that the proposed methods are in favor of others under different rank *** Codes 15A69, 65K05, 65F30, 90C30 Copyright © 2024, The Authors. All rights reserved.

关键词： Tensors

An Adaptive Proximal Inexact Gradient Framework and Its Application to Per-Antenna Constrained Joint Beamforming and Compression Design

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arXiv 2025年

作者： Fan, Xilai Jiang, Bo Liu, Ya-Feng 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 Ministry of Education Key Laboratory of NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing210023 China

In this paper, we propose an adaptive proximal inexact gradient (APIG) framework for solving a class of nonsmooth composite optimization problems involving function and gradient errors. Unlike existing inexact proximal gradient methods, the proposed framework introduces a new line search condition that jointly adapts to function and gradient errors, enabling adaptive stepsize selection while maintaining theoretical guarantees. Specifically, we prove that the proposed framework achieves an ϵ-stationary point within O(ϵ−2) iterations for nonconvex objectives and an ϵ-optimal solution within O(ϵ−1) iterations for convex cases, matching the best-known complexity in this context. We then custom-apply the APIG framework to an important signal processing problem: the joint beamforming and compression problem (JBCP) with per-antenna power constraints (PAPCs) in cooperative cellular networks. This customized application requires careful exploitation of the problem’s special structure such as the tightness of the semidefinite relaxation (SDR) and the differentiability of the dual. Numerical experiments demonstrate the superior performance of our custom-application over state-of-the-art benchmarks for the JBCP. © 2025, CC BY.

关键词： Beamforming

Taking Myopic Best Response Against the Hedge Algorithm 42

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Taking Myopic Best Response Against the Hedge Algorithm

42nd Chinese Control Conference, CCC 2023

作者： Guo, Xinxiang Mu, Yifen School of Mathematical Sciences University of Chinese Academy of Sciences Beijing China Academy of Mathematics and Systems Science The State Key Laboratory of Scientific and Engineering Computing Chinese Academy of Sciences Beijing China Academy of Mathematics and Systems Science The Key Laboratory of Systems and Control Chinese Academy of Sciences Beijing China

ISBN: (纸本)9789887581543

With the rapid development of artificial intelligence (AI), the game between the human and machine/AI becomes more and more common. The related theoretical analysis becomes significant and necessary, which however is still rare. This problem involves evolution analysis and optimal control of dynamic game systems driven by learning algorithms. In this paper, we consider a finitely repeated two-player zero-sum game. We study the dynamic evolution process of the game system, in which one player employs the Hedge algorithm and the other player takes the myopic best response. By the updating formula of the Hedge algorithm, we define a quantity called the state and construct the state Transition Triangle Graph (STTG). Then, we prove that the game system is periodic after o(T) stages. © 2023 Technical Committee on Control Theory, Chinese Association of Automation.

关键词： Artificial intelligence

Precise Analysis of Covariance Identifiability for Activity Detection in Grant-Free Random Access

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arXiv 2024年

作者： Luo, Shengsong Ma, Junjie Xu, Chongbin Wang, Xin the Department of Communication Science and Engineering Fudan University Shanghai200433 China The 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

We consider the identifiability issue of maximum-likelihood based activity detection in massive MIMO-based grant-free random access. An intriguing observation by Chen et al. [1] indicates that the identifiability undergoes a phase transition for commonly-used random user signatures as L2, N and K tend to infinity with fixed ratios, where L, N and K denote the user signature length, the total number of users, and the number of active users, respectively. In this letter, we provide a precise analytical characterization of the phase transition based on a spectral universality conjecture. Numerical results demonstrate excellent agreement between our theoretical predictions and the empirical phase transitions. Copyright © 2024, The Authors. All rights reserved.

关键词： Maximum likelihood

OPTIMIZATION ON PRODUCT MANIFOLDS UNDER A PRECONDITIONED METRIC

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arXiv 2023年

作者： Gao, Bin Peng, Renfeng Yuan, Ya-Xiang State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences China 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

Since optimization on Riemannian manifolds relies on the chosen metric, it is appealing to know that how the performance of a Riemannian optimization method varies with different metrics and how to exquisitely construct a metric such that a method can be accelerated. To this end, we propose a general framework for optimization problems on product manifolds endowed with a preconditioned metric, and we develop Riemannian methods under this metric. Generally, the metric is constructed by an operator that aims to approximate the diagonal blocks of the Riemannian Hessian of the cost function. We propose three specific approaches to design the operator: exact block diagonal preconditioning, left and right preconditioning, and Gauss–Newton type preconditioning. Specifically, we tailor new preconditioned metrics and adapt the proposed Riemannian methods to the canonical correlation analysis and the truncated singular value decomposition problems, which provably accelerate the Riemannian methods. Additionally, we adopt the Gauss–Newton type preconditioning to solve the tensor ring completion problem. Numerical results among these applications verify that a delicate metric does accelerate the Riemannian optimization *** Codes 15A69, 65K05, 65F30, 90C30 Copyright © 2023, The Authors. All rights reserved.

关键词： Cost functions