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Tensor parametric Hamiltonian operator inference

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arXiv 2025年

作者： Vijaywargiya, Arjun McQuarrie, Shane A. Gruber, Anthony Department of Applied and Computational Mathematics and Statistics University of Notre Dame United States Computational Mathematics Center for Computing Research Sandia National Laboratories Scientific Machine Learning Center for Computing Research Sandia National Laboratories

This work presents a tensor-based approach to constructing data-driven reduced-order models corresponding to semi-discrete partial differential equations with canonical Hamiltonian structure. By expressing parameter-varying operators with affine dependence as contractions of a generalized parameter vector against a constant tensor, this method leverages the operator inference framework to capture parametric dependence in the learned reduced-order model via the solution to a convex, least-squares optimization problem. This leads to a concise and straightforward implementation which compactifies previous parametric operator inference approaches and directly extends to learning parametric operators with symmetry constraints—a key feature required for constructing structure-preserving surrogates of Hamiltonian systems. The proposed approach is demonstrated on both a (non-Hamiltonian) heat equation with variable diffusion coefficient as well as a Hamiltonian wave equation with variable wave *** Codes 15A69, 35B30, 65P10 © 2025, CC BY.

关键词： Tensors

ON THE SEPARATING NOETHER NUMBER OF FINITE ABELIAN GROUPS

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arXiv 2025年

作者： Zhao, Kevin Zhong, Qinghai School of Mathematics and Statistics Nanning Normal University Nanning530100 China Center for Applied Mathematics of Guangxi Nanning Normal University Nanning530100 China Institute of Mathematics and Scientific Computing Heinrichstraße 36 Graz8010 Austria

The separating Noether number βsep(G) of a finite group G is the minimal positive integer d such that for every G-module V there is a separating set of degree ≤ d. In this manuscript, we investigate the separating Noether number βsep(G). Among others, we obtain the exact value of βsep(G) for finite abelian groups G, when either G is a p-group or r(G) ∊ {3, 5}.MSC Codes 13A50, 11B75, 20D60 Copyright © 2025, The Authors. All rights reserved.

关键词： Integer linear programming

Learning with Noisy Labels: the Exploration of Error Bounds in Classification

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arXiv 2025年

作者： Liu, Haixia Li, Boxiao Yang, Can Wang, Yang School of Mathematics and Statistics Institute of Interdisciplinary Research for Mathematics and Applied Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan China Department of Mathematics The Hong Kong University of Science and Technology Clear Water Bay China

Numerous studies have shown that label noise can lead to poor generalization performance, negatively affecting classification accuracy. Therefore, understanding the effectiveness of classifiers trained using deep neural networks in the presence of noisy labels is of considerable practical significance. In this paper, we focus on the error bounds of excess risks for classification problems with noisy labels within deep learning frameworks. We begin by exploring loss functions with noise-tolerant properties, ensuring that the empirical minimizer on noisy data aligns with that on the true data. Next, we estimate the error bounds of the excess risks, expressed as a sum of statistical error and approximation error. We estimate the statistical error on a dependent (mixing) sequence, bounding it with the help of the associated independent block sequence. For the approximation error, we first express the classifiers as the composition of the softmax function and a continuous function from [0, 1]d to RK. The main task is then to estimate the approximation error for the continuous function from [0, 1]d to RK. Finally, we focus on the curse of dimensionality based on the low-dimensional manifold assumption. Copyright © 2025, The Authors. All rights reserved.

关键词： Risk assessment

A FOUR-FIELD MIXED FORMULATION FOR INCOMPRESSIBLE FINITE ELASTICITY

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arXiv 2025年

作者： Fu, Guosheng Neunteufel, Michael Schöberl, Joachim Zdunek, Adam Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Fariborz Maseeh Department of Mathematics and Statistics Portland State University 1825 SW Broadway PortlandOR97201 United States Institute of Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 WienA-1040 Austria HB Berrit Solhemsbackarna 73 Spånga SE-163 56 Sweden

In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations [12], involving normal velocity and tangential-normal stress continuous fields, to incompressible finite elasticity. By means of the three-field Hu–Washizu principle, introducing the displacement gradient and 1st Piola–Kirchhoff stress tensor as additional fields, we circumvent the inversion of the constitutive law. We lift the arising distributional derivatives of the displacement gradient to a regular auxiliary displacement gradient field. Static condensation can be applied at the element level, providing a global pure displacement problem to be solved. We present a stabilization motivated by Hybrid Discontinuous Galerkin methods. A solving algorithm is discussed, which asserts the solvability of the arising linearized subproblems for problems with physically positive eigenvalues. The excellent performance of the proposed method is corroborated by several numerical *** Codes 74S05 (Primary) 74B20 (Secondary) Copyright © 2025, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

EHands: Quantum Protocol for Polynomial Computation on Real-Valued Encoded States

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arXiv 2025年

作者： Balewski, Jan Amankwah, Mercy G. Bethel, E. Wes Perciano, Talita Van Beeumen, Roel National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Department of Mathematics Applied Mathematics and Statistics Case Western Reserve University ClevelandOH44106 United States Computer Science Department San Francisco State University San FranciscoCA94132 United States Scientific Data Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States

The novel constructive EHands protocol defines a universal set of quantum operations for multivariable polynomial transformations on quantum processors by introducing four basic subcircuits—multiplication, addition, negation, and parity flip—and by using expectation-value encoding (EVEN) to represent real numbers in quantum states. These elementary arithmetic operations can be systematically composed to compute degree-d polynomials, Pd(x), on a QPU. The resulting quantum circuit structure closely mirrors the stepwise evaluation of polynomials on a classical calculator, providing an intuitive and efficient approach to polynomial computation on quantum hardware. By enabling direct, predictable polynomial and nonlinear data transformations on a QPU, our method reduces dependence on classical post-processing in hybrid quantum-classical algorithms, enabling advancements in many quantum algorithms. The EHands quantum circuits are compact enough to deliver meaningful and accurate results on today’s noisy quantum processors. We present a detailed implementation of P4(x) and report experimental results for polynomial approximations of common functions, obtained using IBM’s Heron-class quantum processors and an ideal Qiskit simulator. Copyright © 2025, The Authors. All rights reserved.

关键词： Polynomial approximation

An automatic approach to develop the fourth-order and L2-stable lattice Boltzmann model for diagonal-anisotropic diffusion equations

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arXiv 2025年

作者： Chen, Ying Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China The State Key Laboratory of Intelligent Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan430074 China

This paper discusses how to develop a high-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the general d(≥ 1)-dimensional diagonal-anisotropic diffusion equation. Such an MRT-LB model considers the transformation matrix constructed in a natural way and the DdQ(2d2 + 1) [(2d2 + 1) discrete velocities in d-dimensional space] lattice structure. A key step in developing the high-order MRT-LB model is to determine the additional adjustable relaxation parameters and weight coefficients, which are used to eliminate the truncation errors at some certain orders of the MRT-LB model, while ensuring the stability of the MRT-LB model. In this work, we first present a unified MRT-LB model for the d-dimensional diagonal-anisotropic diffusion equation. Then, through the direct Taylor expansion, we analyze the macroscopic modified equations of the MRT-LB model up to fourth-order at the diffusive scaling, and further derive the conditions that ensure the MRT-LB model to be fourth-order consistent with the diagonal-anisotropic diffusion equation. In particular, when the diagonal-anisotropic diffusion equation is reduced to the isotropic type, we propose another MRT-LB model with the DdQ(2d+ 1) lattice structure [fewer discrete velocities than the DdQ(2d2 + 1) lattice structure], and the fourth-order conditions are similarly derived. Additionally, we also construct the fourth-order initialization scheme for the present LB method. After that, the condition which guarantees that the MRT-LB model can satisfy the stability structure is explicitly given, and we would like to point out that from a numerical perspective, once the stability structure is satisfied, the MRT-LB model must be L2 stable. In combination with the fourth-order consistent and L2 stability conditions, the relaxation parameters and weight coefficients of the MRT-LB model can be automatically given by a simple computer code. Finally, we perform numerical simulations of several benchmark problems, and f

关键词： Anisotropy

A physics informed neural network approach to simulating ice dynamics governed by the shallow ice approximation

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arXiv 2025年

作者： Chawla, Kapil Holmes, William Department of Mathematics Institute for Scientific Computing and Applied Mathematics Indiana University IN47405 United States Department of Mathematics Cognitive Science Program Indiana University IN47405 United States

In this article we develop a Physics Informed Neural Network (PINN) approach to simulate ice sheet dynamics governed by the Shallow Ice Approximation. This problem takes the form of a time-dependent parabolic obstacle problem [17]. Prior work has used this approach to address the stationary obstacle problem and here we extend it to the time dependent problem [4]. Through comprehensive 1D and 2D simulations, we validate the model’s effectiveness in capturing complex free-boundary conditions. By merging traditional mathematical modeling with cutting-edge deep learning methods, this approach provides a scalable and robust solution for predicting temporal variations in ice thickness. To illustrate this approach in a real world setting, we simulate the dynamics of the Devon Ice Cap, incorporating aerogeophysical data from 2000 [6] and 2018 [22]. © 2025, CC BY.

关键词： Approximation theory

The Silent Majority: Demystifying Memorization Effect in the Presence of Spurious Correlations

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arXiv 2025年

作者： You, Chenyu Dai, Haocheng Min, Yifei Sekhon, Jasjeet S. Joshi, Sarang Duncan, James S. Department of Electrical & Computer Engineering Yale University New HavenCT United States Scientific Computing and Imaging Institute University of Utah Salt Lake CityUT United States Department of Statistics and Data Science Yale University New HavenCT United States Department of Biomedical Engineering Yale University New HavenCT United States Department of Applied Mathematics & Statistics Stony Brook University Stony BrookNY United States Department of Computer Science Stony Brook University Stony BrookNY United States

Machine learning models often rely on simple spurious features - patterns in training data that correlate with targets but are not causally related to them, like image backgrounds in foreground classification. This reliance typically leads to imbalanced test performance across minority and majority groups. In this work, we take a closer look at the fundamental cause of such imbalanced performance through the lens of memorization, which refers to the ability to predict accurately on atypical examples (minority groups) in the training set but failing in achieving the same accuracy in the testing set. This paper systematically shows the ubiquitous existence of spurious features in a small set of neurons within the network, providing the first-ever evidence that memorization may contribute to imbalanced group performance. Through three experimental sources of converging empirical evidence, we find the property of a small subset of neurons or channels in memorizing minority group information. Inspired by these findings, we articulate the hypothesis: the imbalanced group performance is a byproduct of "noisy" spurious memorization confined to a small set of neurons. To further substantiate this hypothesis, we show that eliminating these unnecessary spurious memorization patterns via a novel framework during training can significantly affect the model performance on minority groups. Our experimental results across various architectures and benchmarks offer new insights on how neural networks encode core and spurious knowledge, laying the groundwork for future research in demystifying robustness to spurious correlation. © 2025, CC BY.

关键词： Neurons

Variable Selection for Fixed Effects Varying Coefficient Models

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Acta Mathematica Sinica,English Series 2015年第1期31卷 91-110页

作者： Gao Rong LI Heng LIAN Peng LAI Heng PENG Beijing Center for Scientific and Engineering Computing College of Applied SciencesBeijing University of Technology School of Mathematics and Statistics The University of New South Wales School of Mathematics & Statistics Nanjing University of Information Science & Technology Department of Mathematics Hong Kong Baptist University

We consider the problem of variable selection for the fixed effects varying coefficient models. A variable selection procedure is developed using basis function approximations and group nonconcave penalized functions, and the fixed effects are removed using the proper weight matrices. The proposed procedure simultaneously removes the fixed individual effects, selects the significant variables and estimates the nonzero coefficient functions. With appropriate selection of the tuning parameters, an asymptotic theory for the resulting estimates is established under suitable conditions. Simulation studies are carried out to assess the performance of our proposed method, and a real data set is analyzed for further illustration.

关键词： Varying coefficient model fixed effect variable selection basis function