Tensor network representations of discrete maximum entropy distributions via mean polytopes
本文通过引入计算激活网络(CompActNets)架构，利用凸多面体几何表示离散最大熵分布，解决了在期望约束下如何高效表示这类分布的问题。
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Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domains
This work addresses the challenge of accurately modeling viscous flow in highly porous media using physics-informed neural networks (PINNs), where complex boundaries and fine-scale structures lead to significant accuracy degradation—particularly as pore count increases. To overcome this, the authors propose a novel approach that integrates finite basis PINNs (FBPINNs) with hard boundary constraints. By decomposing the domain into local subregions, the method precisely embeds pore boundary conditions, effectively mitigating spectral bias and non-local effects. Notably, this is the first application of hard constraints within FBPINNs for porous flow problems, circumventing the stiffness and gradient conflicts associated with soft constraints and yielding convergence behavior that is largely independent of pore number. Numerical experiments demonstrate that the proposed framework achieves high accuracy, computational efficiency, and strong scalability in simulating Stokes flow.
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Learning from the Descent Direction: Adaptive Gradient Descent under One-Sided Hölder Regularity
This work proposes an adaptive scalar-step gradient descent method for non-convex optimization that overcomes the restrictive assumptions of traditional approaches. Conventional methods rely on strong regularity conditions—such as global Lipschitz or Hölder continuity of the full gradient—which lead to overly conservative step sizes. In contrast, the proposed algorithm leverages one-sided Hölder regularity to estimate local curvature along the descent direction and dynamically adjusts the step size via a sufficient decrease condition. This strategy relaxes the need for global regularity assumptions while permitting larger steps in flat regions without compromising convergence. Theoretically, the method guarantees optimal stationarity of iterates for non-convex objectives. Empirical results demonstrate superior performance over existing scalar-step gradient methods in binary classification and non-convex Hölder regression tasks, achieving lower final loss, smaller gradient norms, and wider classification margins.
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Minimax Estimation of Kernel Stein Discrepancy: Trace versus Hilbert-Schmidt Scales
This study addresses the minimax optimal estimation of Kernel Stein Discrepancy (KSD) when only the score function of the target distribution and a finite sample are available. By analyzing the spectral properties of the Stein covariance operator, the authors establish for the first time that the minimax risk of KSD estimation is governed by the Hilbert–Schmidt norm of this operator, yielding an optimal convergence rate of √(|C⋆|_HS / n). They propose a positive-part square-root U-statistic estimator that achieves this optimal rate. In contrast, the conventional V-statistic estimator attains only √(tr(C⋆) / n), resulting in an exponentially larger error gap in high-dimensional Gaussian settings, thereby highlighting the substantial advantage of the proposed method.
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MathModDB: A Database for Mathematical Models
Researchers often encounter fragmented information when selecting mathematical models, as formulas, variables, assumptions, and their variants are scattered across disparate literature and domain-specific conventions. To address this challenge, this work proposes and constructs MathModDB—the first open knowledge graph dedicated to mathematical models—built upon the Wikibase framework and grounded in a prior ontological design. MathModDB systematically integrates model metadata, core equations, modeling assumptions, and associated variants. The knowledge graph has been incorporated into the MaRDI portal, forming a synergistic ecosystem with the numerical algorithms knowledge graph MathAlgoDB and the documentation tool MaRDMO. Its practical utility and value are demonstrated through a case study on arc discharge modeling in plasma physics.
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Tensor network representations of discrete maximum entropy distributions via mean polytopes
本文通过引入计算激活网络(CompActNets)架构，利用凸多面体几何表示离散最大熵分布，解决了在期望约束下如何高效表示这类分布的问题。
0 citationsRead paper
Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domains
This work addresses the challenge of accurately modeling viscous flow in highly porous media using physics-informed neural networks (PINNs), where complex boundaries and fine-scale structures lead to significant accuracy degradation—particularly as pore count increases. To overcome this, the authors propose a novel approach that integrates finite basis PINNs (FBPINNs) with hard boundary constraints. By decomposing the domain into local subregions, the method precisely embeds pore boundary conditions, effectively mitigating spectral bias and non-local effects. Notably, this is the first application of hard constraints within FBPINNs for porous flow problems, circumventing the stiffness and gradient conflicts associated with soft constraints and yielding convergence behavior that is largely independent of pore number. Numerical experiments demonstrate that the proposed framework achieves high accuracy, computational efficiency, and strong scalability in simulating Stokes flow.
0 citationsRead paper
Learning from the Descent Direction: Adaptive Gradient Descent under One-Sided Hölder Regularity
This work proposes an adaptive scalar-step gradient descent method for non-convex optimization that overcomes the restrictive assumptions of traditional approaches. Conventional methods rely on strong regularity conditions—such as global Lipschitz or Hölder continuity of the full gradient—which lead to overly conservative step sizes. In contrast, the proposed algorithm leverages one-sided Hölder regularity to estimate local curvature along the descent direction and dynamically adjusts the step size via a sufficient decrease condition. This strategy relaxes the need for global regularity assumptions while permitting larger steps in flat regions without compromising convergence. Theoretically, the method guarantees optimal stationarity of iterates for non-convex objectives. Empirical results demonstrate superior performance over existing scalar-step gradient methods in binary classification and non-convex Hölder regression tasks, achieving lower final loss, smaller gradient norms, and wider classification margins.
0 citationsRead paper
Minimax Estimation of Kernel Stein Discrepancy: Trace versus Hilbert-Schmidt Scales
This study addresses the minimax optimal estimation of Kernel Stein Discrepancy (KSD) when only the score function of the target distribution and a finite sample are available. By analyzing the spectral properties of the Stein covariance operator, the authors establish for the first time that the minimax risk of KSD estimation is governed by the Hilbert–Schmidt norm of this operator, yielding an optimal convergence rate of √(|C⋆|_HS / n). They propose a positive-part square-root U-statistic estimator that achieves this optimal rate. In contrast, the conventional V-statistic estimator attains only √(tr(C⋆) / n), resulting in an exponentially larger error gap in high-dimensional Gaussian settings, thereby highlighting the substantial advantage of the proposed method.
0 citationsRead paper
MathModDB: A Database for Mathematical Models
Researchers often encounter fragmented information when selecting mathematical models, as formulas, variables, assumptions, and their variants are scattered across disparate literature and domain-specific conventions. To address this challenge, this work proposes and constructs MathModDB—the first open knowledge graph dedicated to mathematical models—built upon the Wikibase framework and grounded in a prior ontological design. MathModDB systematically integrates model metadata, core equations, modeling assumptions, and associated variants. The knowledge graph has been incorporated into the MaRDI portal, forming a synergistic ecosystem with the numerical algorithms knowledge graph MathAlgoDB and the documentation tool MaRDMO. Its practical utility and value are demonstrated through a case study on arc discharge modeling in plasma physics.
0 citationsRead paper