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Σ-Attention: A Transformer-based operator learning framework for self-energy in strongly correlated systems

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arXiv 2025年

作者： Zhu, Yuanran Rosenberg, Peter Huang, Zhen Bassi, Hardeep Yang, Chao Zhang, Shiwei Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory Berkeley94720 United States Center for Computational Quantum Physics Flatiron Institute 162 5th Avenue New YorkNY10010 United States Department of Mathematics University of California BerkeleyCA94720 United States Department of Applied Mathematics University of California MercedCA95343 United States

We introduce Σ-Attention, a Transformer-based operator-learning framework to address a key computational challenge in correlated materials. Our approach utilizes an Encoder-Only Transformer as an ansatz to approximate the self-energy operator of strongly correlated electronic systems. By creating a batched dataset that combines results from three complementary approaches: many-body perturbation theory, strong-coupling expansion, and exact diagonalization, each effective in specific parameter regimes, Σ-Attention is applied to learn a universal approximation for the self-energy operator that is valid across all regimes. This hybrid strategy leverages the strengths of existing methods while relying on the transformer’s ability to generalize beyond individual limitations. More importantly, the scalability of the Transformer architecture allows the learned self-energy to be extended to systems with larger sizes, leading to much improved computational scaling. Using the 1D Hubbard model, we demonstrate that Σ-Attention can accurately predict the Matsubara Green’s function and hence effectively captures the Mott transition at finite temperature. Our framework offers a promising and scalable pathway for studying strongly correlated systems with many possible generalizations. Copyright © 2025, The Authors. All rights reserved.

关键词： Perturbation techniques

Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem

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arXiv 2025年

作者： Elman, Howard C. Liang, Jiaxing Sánchez-Vizuet, Tonatiuh Department of Computer Science Institute for Advanced Computer Studies University of Maryland College Park United States Department of Computational Applied Mathematics & Operations Research Rice University United States Department of Mathematics The University of Arizona United States

We explore a hybrid technique to quantify the variability in the numerical solutions to a free boundary problem associated with magnetic equilibrium in axisymmetric fusion reactors amidst parameter uncertainties. The method aims at reducing computational costs by integrating a surrogate model into a multilevel Monte Carlo method. The resulting surrogate-enhanced multilevel Monte Carlo methods reduce the cost of simulation by factors as large as 104 compared to standard Monte Carlo simulations involving direct numerical solutions of the associated Grad-Shafranov partial differential equation. Accuracy assessments also show that surrogate-based sampling closely aligns with the results of direct computation, confirming its effectiveness in capturing the behavior of plasma boundary and geometric *** Codes 65Z05, 65C05, 62P35, 35R35, 35R60 Copyright © 2025, The Authors. All rights reserved.

关键词： Stochastic systems

Sharp concentration of simple random tensors

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arXiv 2025年

作者： Al-Ghattas, Omar Chen, Jiaheng Sanz-Alonso, Daniel Department of Statistics University of Chicago IL60637 United States Committee on Computational and Applied Mathematics University of Chicago IL60637 United States

This paper establishes sharp dimension-free concentration inequalities and expectation bounds for the deviation of the sum of simple random tensors from its expectation. As part of our analysis, we use generic chaining techniques to obtain a sharp high-probability upper bound on the suprema of multi-product empirical processes. In so doing, we generalize classical results for quadratic and product empirical processes to higher-order settings. © 2025, CC BY.

关键词： Tensors

ENHANCED GRADIENT RECOVERY-BASED A POSTERIORI ERROR ESTIMATOR AND ADAPTIVE FINITE ELEMENT METHOD FOR ELLIPTIC EQUATIONS

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arXiv 2025年

作者： Liu, Ying Xiao, Jingjing Yi, Nianyu Cao, Huihui Department of Mathematics School of Science Xi’an University of Technology Xi’an710000 China School of Mathematics Shandong University Jinan250100 China School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan National Center for Applied Mathematics Xiangtan411105 China

Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators, one can not establish upper and lower a posteriori error bounds for the classical recovery type error estimators without the saturation assumption. In this paper, we first present three examples to show the unsatisfactory performance in the practice of standard residual or recovery-type error estimators, then, an improved gradient recovery-based a posteriori error estimator is constructed. The proposed error estimator contains two parts, one is the difference between the direct and post-processed gradient approximations, and the other is the residual of the recovered gradient. The reliability and efficiency of the enhanced estimator are derived. Based on the improved recovery-based error estimator and the newest-vertex bisection refinement method with a tailored mark strategy, an adaptive finite element algorithm is designed. We then prove the convergence of the adaptive method by establishing the contraction of gradient error plus oscillation. Numerical experiments are provided to illustrate the asymptotic exactness of the new recovery-based a posteriori error estimator and the high efficiency of the corresponding adaptive algorithm. Copyright © 2025, The Authors. All rights reserved.

关键词： Finite element method

The effect of external magnetic field on projectile in the stopping power of warm dense deuterium for electrons

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The European Physical Journal Special Topics 2025年 1-16页

作者： Chen, Yuzhu Mo, Chongjie Xue, Quanxi Fu, Zhenguo Liu, Hao Department of Applied Physics School of Physics and Electronics Hunan University Changsha China Beijing Computational Science Research Center Beijing China State Key Laboratory of Intense Pulsed Radiation Simulation and Effect Northwest Institute of Nuclear Technology Xi’an China Institute of Applied Physics and Computational Mathematics Beijing China

The stopping power of warm dense plasmas for electrons is a critical aspect in the study of hot electron transport. An externally applied strong magnetic field can significantly influence electron transport behavior due to various factors. However, the impact of external magnetic fields on the motion of incident particles is often overlooked. Through molecular dynamics simulations using the electron force field (eFF) method, this study investigates the stopping process of individual hot electrons in warm dense deuterium plasma under an applied longitudinal magnetic field. Results show that, at typical laboratory magnetic field intensities, the magnetic field significantly alters electron trajectories without notable effects on average stopping power, trajectory length, or scattering angle. Even with increased magnetic field intensity beyond 500 kT, it doesn’t affect the total kinetic energy loss of incident electrons but reduces stopping power by compressing the scattering angle distribution width. Due to the increase in the scattering angle distribution width with intensified fluctuations in high-temperature targets, the impact of the additional magnetic field on stopping power becomes more pronounced with an increase in target temperature.

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Is a single lethal electric field threshold sufficient to characterize the lesion size in computational modeling of cardiac pulsed-field ablation?

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Heart Rhythm O2

Heart Rhythm O2 2025年第5期6卷 671-677.0页

作者： Petras, Argyrios Amoros Figueras, Gerard Moreno Weidmann, Zoraida García-Sánchez, Tomás Viladés Medel, David Ivorra, Antoni Guerra, Jose M. Gerardo-Giorda, Luca RICAM–Johann Radon Institute for Computational and Applied Mathematics Linz Austria Department of Cardiology Hospital de la Santa Creu i Sant Pau Universitat Autònoma de Barcelona CIBER CV Barcelona Spain Department of Information and Communication Technologies Universitat Pompeu Fabra Barcelona Spain Institute for Mathematical Methods in Medicine and Data-Based Modelling Johannes Kepler University Linz Austria

Background: Pulsed-field ablation (PFA) is a novel cardiac ablation technology based on irreversible electroporation (IRE). PFA computational models rely on identification of a lethal electric field threshold to predict the IRE area. However, the predicted lesion anisotropy ratios (width over depth) vary extensively among recent studies, and these discrepancies remain a subject of discussion. Objective: This work aims to evaluate the predicted lesion anisotropy ratios using a PFA computational model by applying it to an open-chest in vivo porcine model geometry. Methods: Six domestic swine underwent epicardial PFA applications using a previously described waveform protocol. Animals were killed at least 3 hours after the last ablation, and lesions were assessed using triphenyltetrazolium chloride (TTC) staining. Numeric simulations were performed on a segmented and meshed porcine thoracic computed tomography (CT) scan, mimicking the open-chest experimental setup. Results: The maximum width of all simulated lesions was observed at the epicardial surface. The anisotropy ratios (AR) of the experimental lesions were smaller than the simulated ones (AR experimental vs simulated, 1.0–1.7 vs 2–2.7;Q1–Q3 quartiles). Increasing the peak voltage resulted in larger lesions;however, the computational model clearly underestimated the increase in lesion depth compared with the experimental data. Conclusion: Our computational model shows that a single lethal electric field threshold is insufficient to accurately predict both lesion depth and width in cardiac PFA. Our study suggests that for the given PFA waveforms, a threshold between 270 and 500 V/cm provides satisfactory lesion depth estimations, and a higher threshold between 790 and 1000 V/cm better captures the lesion width. © 2025 Heart Rhythm Society

关键词： Cardiac ablation Computer simulations Electric field threshold Mathematical modeling Pulsed field ablation

Accuracy Versus Predominance: Reassessing the Validity of the Quasi-Steady-State Approximation

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Bulletin of Mathematical Biology 2025年第6期87卷 1-32页

作者： Srivastava, Kashvi Eilertsen, Justin Booth, Victoria Schnell, Santiago Department of Mathematics University of Michigan Ann Arbor 48109 MI United States Mathematical Reviews American Mathematical Society 416 4th Street Ann Arbor 48103 MI United States Department of Biological Sciences and Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame 46556 IN United States

The application of the standard quasi-steady-state approximation to the Michaelis-Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, e0, to the Michaelis constant, KM, decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while e0/KM provides an indication of the standard quasi-steady-state approximation’s asymptotic accuracy, the standard quasi-steady-state approximation’s predominance relies on a small ratio of e0 to the Van Slyke-Cullen constant, K. Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of e0/K offers the most accurate measure of the validity of the standard quasi-steady-state approximation. © The Author(s) 2025.

关键词： Michaelis–Menten reaction mechanism Quasi-steady-state approximation Singular perturbation Timescale separation

Stability and Time-Step Constraints of Exponential Time Differencing Runge–Kutta Discontinuous Galerkin Methods for Advection-Diffusion Equations

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arXiv 2025年

作者： Xu, Ziyao Sun, Zheng Zhang, Yong-Tao Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Department of Mathemtatics The University of Alabama TuscaloosaAL35487 United States

In this paper, we investigate the stability and time-step constraints for solving advection-diffusion equations using exponential time differencing (ETD) Runge–Kutta (RK) methods in time and discontinuous Galerkin (DG) methods in space. We demonstrate that the resulting fully discrete scheme is stable when the time-step size is upper bounded by a constant. More specifically, when central fluxes are used for the advection term, the schemes are stable under the time-step constraint τ ≤ τ0d/a2, while when upwind fluxes are used, the schemes are stable if τ ≤ max {τ0d/a2, c0h/a}. Here, τ is the time-step size, h is the spatial mesh size, and a and d are constants for the advection and diffusion coefficients, respectively. The constant c0 is the CFL constant for the explicit RK method for the purely advection equation, and τ0 is a constant that depends on the order of the ETD-RK method. These stability conditions are consistent with those of the implicit-explicit RKDG method. The time-step constraints are rigorously proved for the lowest-order case and are validated through Fourier analysis for higher-order cases. Notably, the constant τ0 in the fully discrete ETD-RKDG schemes appears to be determined by the stability condition of their semidiscrete (continuous in space, discrete in time) ETD-RK counterparts and is insensitive to the polynomial degree and the specific choice of the DG method. Numerical examples, including problems with nonlinear convection in one and two dimensions, are provided to validate our findings. Copyright © 2025, The Authors. All rights reserved.

关键词： Galerkin methods

Multicontinuum Modeling of Time-Fractional Diffusion-Wave Equation in Heterogeneous Media

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arXiv 2025年

作者： Bai, Huiran Ammosov, Dmitry Yang, Yin Xie, Wei Kobaisi, Mohammed Al School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan Hunan Xiangtan411105 China Chemical and Petroleum Engineering Department Khalifa University of Science and Technology Abu Dhabi127788 United Arab Emirates

This paper considers a time-fractional diffusion-wave equation with a high-contrast heterogeneous diffusion coefficient. A numerical solution to this problem can present great computational challenges due to its multiscale nature. Therefore, in this paper, we derive a multicontinuum time-fractional diffusion-wave model using the multicontinuum homogenization method. For this purpose, we formulate constraint cell problems considering various homogenized effects. These cell problems are implemented in oversampled regions to avoid boundary effects. By solving the cell problems, we obtain multicontinuum expansions of fine-scale solutions. Then, using these multicontinuum expansions and supposing the smoothness of the macroscopic variables, we rigorously derive the corresponding multicontinuum model. Finally, we present numerical results for two-dimensional model problems with different time-fractional derivatives to verify the accuracy of our proposed *** Codes 35B27 (Primary) 26A33, 65M60 (Secondary) Copyright © 2025, The Authors. All rights reserved.

关键词： Wave equations