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Exploring biologically inspired mechanisms of adversarial robustness

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Neural computing and Applications 2025年 1-12页

作者： Holzhausen, Konstantin Merlid, Mia Torvik, Håkon Olav Malthe-Sørenssen, Anders Lepperød, Mikkel Elle Department of Physics University of Oslo Sem Sælands vei 24 Fysikkbygningen Oslo0371 Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway

Backpropagation-optimized artificial neural networks, while precise, lack robustness, leading to unforeseen behaviors that affect their safety. Biological neural systems do solve some of these issues already. Unlike artificial models, biological neurons adjust connectivity based on neighboring cell activity. Understanding the biological mechanisms of robustness can pave the way toward building trustworthy and safe systems. Robustness in neural representations is hypothesized to correlate with the smoothness of the encoding manifold. Recent work suggests power law covariance spectra, which were observed studying the primary visual cortex of mice, to be indicative of a balanced trade-off between accuracy and robustness in representations. Here, we show that unsupervised local learning models with winner takes all dynamics learn such power law representations, providing upcoming studies a mechanistic model with that characteristic. Our research aims to understand the interplay between geometry, spectral properties, robustness, and expressivity in neural representations. Hence, we study the link between representation smoothness and spectrum by using weight, Jacobian and spectral regularization while assessing performance and adversarial robustness. Our work serves as a foundation for future research into the mechanisms underlying power law spectra and optimally smooth encodings in both biological and artificial systems. The insights gained may elucidate the mechanisms that realize robust neural networks in mammalian brains and inform the development of more stable and reliable artificial systems. © The Author(s) 2025.

关键词： Generative adversarial networks

Correction: Exploring biologically inspired mechanisms of adversarial robustness(Neural computing and Applications, 10.1007/s00521-025-11019-6)

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Neural computing and Applications 2025年 1-2页

In this article, panel (b) of Fig. 6 was inadvertently printed in black and white instead of colour;the figure should have appeared as shown below. (Figure presented.) Norm of the Jacobian of latent activations h^ (upper panel) and decision landscape (lower panel) of studied models. Similarly to [12], the displayed planes are created by grid sampling from a random plane projection in the model’s input space The original article has been corrected. © The Author(s) 2025.

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An energy stable and conservative multiplicative dynamical low-rank discretization for the Su-Olson problem

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arXiv 2025年

作者： Baumann, Lena Einkemmer, Lukas Klingenberg, Christian Kusch, Jonas University of Wuerzburg Department of Mathematics Wuerzburg Germany University of Innsbruck Numerical Analysis and Scientific Computing Innsbruck Austria Norwegian University of Life Sciences Department of Data Science Ås Norway

computing numerical solutions of the thermal radiative transfer equations on a finely resolved grid can be costly due to high computational and memory requirements. A numerical reduced order method that has recently been applied to a wide variety of kinetic partial differential equations is the concept of dynamical low-rank approximation (DLRA). In this paper, we consider the thermal radiative transfer equations with Su-Olson closure, leading to a linearized kinetic model. For the conducted theoretical and practical considerations we use a multiplicative splitting of the distribution function that poses additional challenges in finding an energy stable discretization and deriving a hyperbolic Courant-Friedrichs-Lewy (CFL) condition. We propose such an energy stable DLRA scheme that makes use of the augmented basis update & Galerkin integrator. This integrator allows for additional basis augmentations, enabling us to give a mathematically rigorous proof of energy stability and local mass conservation. numerical examples confirm the derived properties and show the computational advantages of the DLRA scheme compared to a numerical solution of the full system of *** Codes 35L65, 35Q49, 65M12, 65M22 © 2025, CC BY-NC-ND.

关键词： Radiative transfer

Hexagons all the way down: grid cells as a conformal isometric map of space

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PLoS Computational Biology 2025年第2期21卷 e1012804页

作者： Schøyen, Vemund Sigmundson Beshkov, Kosio Pettersen, Markus Borud Hermansen, Erik Holzhausen, Konstantin Malthe-Sørenssen, Anders Fyhn, Marianne Lepperød, Mikkel Elle Department of Biosciences University of Oslo Oslo Norway Department of Physics University of Oslo Oslo Norway Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory AS Oslo Norway

Grid cells in the entorhinal cortex are known for their hexagonal spatial activity patterns and are thought to provide a neural metric for space, and support path integration. In this study, we further investigate grid cells as a metric of space by optimising them for a conformal isometric (CI) map of space using a model based on a superposition of plane waves. By optimising the phases within a single grid cell module, we find that the module can form a CI of two-dimensional flat space with phases arranging into a regular hexagonal pattern, supporting an accurate spatial metric. Additionally, we find that experimentally recorded grid cells exhibit CI properties, with one example module showing a phase arrangement similar to the hexagonal pattern observed in our model. These findings provide computational and preliminary experimental support for grid cells as a CI-based spatial representation. We also examine other properties that emerge in CI-optimised modules, including consistent energy expenditure across space and the minimal cell count required to support unique representation of space and maximally topologically persistent toroidal population activity. Altogether, our results suggest that grid cells are well-suited to form a CI map, with several beneficial properties arising from this organisation. © 2025 Schøyen et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

关键词： Cytology

PARAMETER-ROBUST PRECONDITIONERS FOR A FOUR-FIELD THERMO-POROELASTICITY MODEL

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arXiv 2025年

作者： Cai, Mingchao Kuchta, Miroslav Li, Jingzhi Li, Ziliang Mardal, Kent-Andre Department of Mathematics Morgan State University BaltimoreMD21251 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Department of Mathematics Southern University of Science and Technology Shenzhen518055 China

We study a thermo-poroelasticity model which describes the interaction between the deformation of an elastic porous material and fluid flow under non-isothermal conditions. The model involves several parameters that can vary significantly in practical applications, posing a challenge for developing discretization techniques and solution algorithms that handle such variations effectively. We propose a four-field formulation and apply a conforming finite element discretization. The primary focus is on constructing and analyzing preconditioners for the resulting linear system. Two preconditioners are proposed: one involves regrouping variables and treating the 4-by-4 system as a 2-by-2 block form, while the other is directly constructed from the 4-by-4 coupled operator. Both preconditioners are demonstrated to be robust with respect to variations in parameters and mesh refinement. numerical experiments are presented to demonstrate the effectiveness of the proposed preconditioners and validate their theoretical performance under varying parameter *** Codes 65M60, 65F08, 65F10 Copyright © 2025, The Authors. All rights reserved.

关键词： Flow of fluids

PARAMETER-ROBUST PRECONDITIONER FOR STOKES-DARCY COUPLED PROBLEM WITH LAGRANGE MULTIPLIER

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arXiv 2025年

作者： Hu, Xiaozhe Kuchta, Miroslav Mardal, Kent-Andre Wang, Xue Department of Mathematics Tufts University MedfordMA02155 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Simula Research Laboratory Oslo0164 Norway Department of Mathematics University of Oslo Blindern Oslo0316 Norway School of Mathematics Shandong University Jinan250100 China

In this paper, we propose a parameter-robust preconditioner for the coupled Stokes-Darcy problem equipped with various boundary conditions, enforcing the mass conservation at the interface via a Lagrange multiplier. We rigorously establish that the coupled system is well-posed with respect to physical parameters and mesh size and provide a framework for constructing parameter-robust preconditioners. Furthermore, we analyze the convergence behavior of the Minimal Residual method in the presence of small outlier eigenvalues linked to specific boundary conditions, which can lead to slow convergence or stagnation. To address this issue, we employ deflation techniques to accelerate the convergence. Finally, numerical experiments confirm the effectiveness and robustness of the proposed approach. Copyright © 2025, The Authors. All rights reserved.

关键词： Lagrange multipliers

THE LATENT VARIABLE PROXIMAL POINT ALGORITHM FOR VARIATIONAL PROBLEMS WITH INEQUALITY CONSTRAINTS

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arXiv 2025年

作者： Dokken, Jørgen S. Farrell, Patrick E. Keith, Brendan Papadopoulos, Ioannis P.A. Surowiec, Thomas M. Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Oslo Norway Mathematical Institute University of Oxford Oxford United Kingdom Mathematical Institute Faculty of Mathematics and Physics Charles University Prague Czech Republic Division of Applied Mathematics Brown University ProvidenceRI United States Weierstrass Institute Berlin Germany

The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides;in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge–Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. In this paper, we describe the LVPP algorithm in a general form and apply it to twelve problems from across mathematics. © 2025, CC BY.

关键词： Consensus algorithm

Natural convection in the horizontal annulus: critical Rayleigh number for the steady problem

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arXiv 2025年

作者： Passerini, Arianna Rummler, Bernd Růžička, Michael Thäter, Gudrun Department of Mathematics and Scientific Computing Università di Ferrara Via Saragat 1 Ferrara44100 Italy Institute for Analysis and Numerics Otto-von-Guericke-Universität Magdeburg PF 4120 Magdeburg39016 Germany Institute for Applied Mathematics Universität Freiburg Eckerstr. 1 Freiburg79104 Germany Institute for Applied & Numerical Mathematics KIT Karlsruhe76128 Germany

For the 2D Oberbeck-Boussinesq system in an annulus we are looking for the critical Rayleigh number for which the (nonzero) basic flow loses stability. For this we consider the corresponding Euler-Lagrange equations and construct a precise functional analytical frame for the Laplace- and the Stokes problem as well as the Bilaplacian operator in this domain. With this frame and the right set of basis functions it is then possible to construct and apply a numerical scheme providing the critical Rayleigh number. These results will be published in ZAMM in *** Codes 76E09, 76E06, 76D03, 35Q35 © 2025, CC BY.

关键词： Rayleigh number

Algebraic multigrid methods based on element preconditioning

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2001年第4期78卷 575-598页

作者： Haase, G Langer, U Reitzinger, S Schöberl, J Institute for Analysis and Computational Mathematics Department of Computational Mathematics and Optimization Special Research Program (SFB) F013 Numerical and Symbolic Scientific Computing

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stuben's method. Ruge/Stuben's algorithm is robust for M-matrices, but unfortunately the "region of robustness" between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy. For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem. The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.

关键词： finite element equations algebraic multigrid element preconditioning technique preconditioned conjugate gradient method