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SEMIALGEBRAIC NEURAL NETWORKS: FROM ROOTS TO REPRESENTATIONS

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arXiv 2025年

作者： Mis, S. David Lassas, Matti de Hoop, Maarten V. Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX77005 United States Department of Mathematics and Statistics University of Helsinki P.O. Box 68 HelsinkiFI-00014 Finland Simons Chair in Computational Applied Mathematics and Earth Science Rice University HoustonTX77005 United States

Many numerical algorithms in scientific computing—particularly in areas like numerical linear algebra, PDE simulation, and inverse problems—produce outputs that can be represented by semialgebraic functions;that is, the graph of the computed function can be described by finitely many polynomial equalities and inequalities. In this work, we introduce Semialgebraic Neural Networks (SANNs), a neural network architecture capable of representing any bounded semialgebraic function, and computing such functions up to the accuracy of a numerical ODE solver chosen by the programmer. Conceptually, we encode the graph of the learned function as the kernel of a piecewise polynomial selected from a class of functions whose roots can be evaluated using a particular homotopy continuation method. We show by construction that the SANN architecture is able to execute this continuation method, thus evaluating the learned semialgebraic function. Furthermore, the architecture can exactly represent even discontinuous semialgebraic functions by executing a continuation method on each connected component of the target function. Lastly, we provide example applications of these networks and show they can be trained with traditional deep-learning techniques. © 2025, CC BY.

关键词： Inverse problems

Integral equations for flexural-gravity waves: analysis and numerical methods

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arXiv 2025年

作者： Askham, Travis Hoskins, Jeremy G. Nekrasov, Peter Rachh, Manas Department of Mathematical Sciences New Jersey Institute of Technology NewarkNJ07102 United States Department of Statistics and CCAM University of Chicago ChicagoIL60637 United States Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States

In this work, we develop a fast and accurate method for the scattering of flexural-gravity waves by a thin plate of varying thickness overlying a fluid of infinite depth. This problem commonly arises in the study of sea ice and ice shelves, which can have complicated heterogeneities that include ridges and rolls. With certain natural assumptions on the thickness, we present an integral equation formulation for solving this class of problems and analyze its mathematical properties. The integral equation is then discretized and solved using a high-order-accurate, FFT-accelerated algorithm. The speed, accuracy, and scalability of this approach are demonstrated through a variety of illustrative examples. © 2025, CC BY.

关键词： Gravity waves

Solving Inverse Problems with Deep Linear Neural Networks: Global Convergence Guarantees for Gradient Descent with Weight Decay

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arXiv 2025年

作者： Laus, Hannah Parkinson, Suzanna Charisopoulos, Vasileios Krahmer, Felix Willett, Rebecca Department of Mathematics Technical University of Munich Germany Germany Committee on Computational and Applied Mathematics University of Chicago United States Data Science Institute University of Chicago United States Technical University of Munich Germany Department of Computer Science University of Chicago United States Department of Statistics University of Chicago United States

Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few measurements generated via a known acquisition procedure. In particular, neural networks perform well empirically but have limited theoretical guarantees. In this work, we study an underdetermined linear inverse problem that admits several possible solution mappings. A standard remedy (e.g., in compressed sensing) establishing uniqueness of the solution mapping is to assume knowledge of latent low-dimensional structure in the source signal. We ask the following question: do deep neural networks adapt to this low-dimensional structure when trained by gradient descent with weight decay regularization? We prove that mildly overparameterized deep linear networks trained in this manner converge to an approximate solution that accurately solves the inverse problem while implicitly encoding latent subspace structure. To our knowledge, this is the first result to rigorously show that deep linear networks trained with weight decay automatically adapt to latent subspace structure in the data under practical stepsize and weight initialization schemes. Our work highlights that regularization and overparameterization improve generalization, while overparameterization also accelerates convergence during training. © 2025, CC BY.

关键词： Deep neural networks

THE ELASTIC RAY TRANSFORM

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arXiv 2025年

作者： Ilmavirta, Joonas Kykkänen, Antti Saksala, Teemu Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX United States Department of Mathematics NC State University RaleighNC United States

We introduce and study a new family of tensor tomography problems. At rank 2 it corresponds to linearization of travel time of elastic waves, measured for all polarizations. We provide a kernel characterization for ranks up to 2. The kernels consist of potential tensors, but in an unusual sense: the associated differential operators have degree 2 instead of the familiar 1. The proofs are based on Fourier analysis, Helmholtz decompositions, and cohomology. Copyright © 2025, The Authors. All rights reserved.

关键词： Tensors

Independence, induced subgraphs, and domination in K1,r-free graphs

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arXiv 2025年

作者： Caro, Yair Davila, Randy Henning, Michael A. Pepper, Ryan Department of Mathematics University of Haifa–Oranim Tivon36006 Israel Department of Computational Applied Mathematics & Operations Research Rice University HoustonTX77005 United States Department of Mathematics and Applied Mathematics University of Johannesburg Auckland Park2006 South Africa Department of Mathematics and Statistics University of Houston–Downtown HoustonTX77002 United States

Let G be a graph and F a family of graphs. Define αF(G) as the maximum order of any induced subgraph of G that belongs to the family F. For the family F of graphs with chromatic number at most k, we prove that if G is K1,r-free, then αF(G) ≤ (r − 1)kγ(G), where γ(G) is the domination number. When F is the family of empty graphs, this bound simplifies to α(G) ≤ 2γ(G) for K1,3free (claw-free) graphs, where α(G) is the independence number of G. For d-regular graphs, this is further refined to the bound α(G) ≤ 2 (d+1/d+2) γ(G), which is tight for d ∈ {2, 3, 4}. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for K1,r-free graphs, we have αF(G) ≤ r(Kr, F∗)γ(G), where F∗ is the set of graphs not in F. Specializing to Kq-free graphs, we show αF(G) ≤ (r(Kq, Kr) − 1)γ(G). Finally, for the k-independence number αk(G), we prove that if G is K1,r-free with order n and minimum degree δ ≥ k + 1, (formula presented) and this bound is sharp for all parameters. Copyright © 2025, The Authors. All rights reserved.

关键词： Graph theory

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

The Fokas method for evolution partial differential equations

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Partial Differential Equations in applied mathematics 2025年 14卷

作者： Chatziafratis, A. Fokas, A.S. Kalimeris, K. Department of Mathematics and Statistics School of Pure and Applied Sciences University of Cyprus Cyprus Institute of Applied and Computational Mathematics Foundation for Research and Technology Crete Greece Department of Applied Mathematics and Theoretical Physics University of Cambridge United Kingdom Mathematics Research Center Academy of Athens Greece

In the late 1990s a novel methodology was introduced for solving boundary value problems for linear and integrable nonlinear PDEs. This new approach is known as the Unified Transform or the Fokas method. Here we discuss important developments regarding the implementation of this methodology to evolution PDEs. In particular, we analyse linear PDEs, including the heat equation, the Sobolev–Barenblatt pseudo-parabolic model, the Rubinshtein–Aifantis double-diffusion system, and the linearized Cahn–Hilliard model, as well as certain integrable nonlinear PDEs, like the nonlinear Schrödinger equation. Regarding the latter equation, emphasis is placed on the so-called linearizable boundary conditions, which remarkably, include the well-studied x-periodic initial value problem. © 2025 The Authors

关键词： Evolution PDEs Fokas method Integrable PDEs Unified transform

Advancements on multi-fidelity random Fourier neural networks: application to hurricane modeling for offshore wind energy

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Advancements on multi-fidelity random Fourier neural network...

AIAA Science and Technology Forum and Exposition, AIAA SciTech Forum 2025

作者： Davis, Owen Geraci, Gianluca Wentz, Jacqueline M. King, Ryan N. Cortiella, Alexandre Rybchuk, Alex Gomez, Miguel Sanchez Deskos, Georgios Motamed, Mohammad Optimization and Uncertainty Quantification Sandia National Laboratories AlbuquerqueNM United States Computational Data Science Sandia National Laboratories LivermoreCA United States National Wind Technology Center National Renewable Energy Laboratory GoldenCO United States Department of Applied Mathematics and Statistics University of New Mexico AlbuquerqueNM United States

ISBN: (数字)9781624107238

ISBN: (纸本)9781624107238

Multi-fidelity approaches are emerging as effective strategies in computational science to handle otherwise intractable tasks like Uncertainty Quantification (UQ), training of Machine Learning (ML) models, and optimization, for expensive high-fidelity applications in which the amount of available simulations or data is limited. The main idea is simple: large datasets generated for low-fidelity approximations of the problem at hand are fused with a much sparser dataset for the target (high-fidelity) system. In this paper, we build on our recent success in designing random Fourier Neural Networks (rFNNs) [1] to target problems arising in wind energy applications and in particular problems of interest for hurricane modeling. In this context, data for the high-fidelity models are limited and lower fidelity alternatives are needed. In this work, we introduce a novel multi-fidelity training approach for our rFNNs and demonstrate its use on a simple verification problem and on a hurricane modeling problem in which high-fidelity data are generated via Large-Eddy Simulations (LES), while low-fidelity data are given by a mesoscale model. Initial results demonstrate how the multi-fidelity training approach can improve the quality of the resulting surrogate. © 2025, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.

关键词： Large eddy simulation

Asymptotic normality of the Conditional Value-at-Risk based Pickands estimator

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statistics and Probability Letters 2025年 223卷

作者： Li, Yizhou Polak, Paweł Department of Applied Mathematics and Statistics Stony Brook University United States Institute for Advanced Computational Science Stony Brook University United States

We show weak convergence of the empirical Conditional Value-at-Risk (CVaR) in functional space and the asymptotic normality of the CVaR-based Pickands estimator from Chen (2021). These results demonstrate that the CVaR-based estimator has significantly lower asymptotic variance than analogous VaR-based constructions. © 2025 Elsevier B.V.

关键词： Asymptotic normality Conditional Value-at-Risk Extreme value index Pickands estimator Second-order regular variation