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

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

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FULLY DISCRETE analysis OF THE GALERKIN POD NEURAL NETWORK APPROXIMATION WITH APPLICATION TO 3D ACOUSTIC WAVE SCATTERING

arXiv

引用

arXiv 2025年

作者： Dölz, Jürgen Henríquez, Fernando Institute for Numerical Simulation University of Bonn Friedrich-Hirzebruch-Allee 7 Bonn53115 Germany Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrae 8-10 WienA-1040 Austria

In this work, we consider the approximation of parametric maps using the so-called Galerkin POD-NN method. This technique combines the computation of a reduced basis via proper orthogonal decomposition (POD) and artificial neural networks (NNs) for the construction of fast surrogates of said parametric maps. We provide a fully discrete error analysis of this method that accounts for different discretization errors, including the number of reduced basis in the approximation of the solution manifold, truncation in the parameter space, and, most importantly, the number of samples in the computation of the reduced space, together with the effect of the use of NNs in the approximation of the reduced coefficients. Following this error analysis we provide a-priori bounds on the required POD tolerance, the resulting POD ranks, and NN-parameters to maintain the order of convergence of quasi Monte Carlo sampling techniques. The main advantage of Galerkin POD-NN over existing traditional techniques is that the online and offline phases of the projection-based reduced basis method are completely decoupled, making the evaluation of the constructed surrogate in run-time considerably faster without the need for an hyper-reduction technique. We conclude this work by showcasing the applicability of this method through a practical industrial application: the sound-soft acoustic scattering problem by a parametrically defined scatterer in three physical dimensions. Copyright © 2025, The Authors. All rights reserved.

关键词： Monte Carlo methods

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Conservation Properties of the Augmented Basis Update & Galerkin Integrator for Kinetic Problems

SSRN

引用

SSRN 2023年

作者： Einkemmer, Lukas Daniel Kusch, Jonas Schotthöfer, Steffen University of Innsbruck Numerical Analysis and Scientific Computing Innsbruck Austria Norwegian University of Life Sciences Scientific Computing Ås Norway Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

numerical simulations of kinetic problems can become prohibitively expensive due to their large memory footprint and computational costs. A method that has proven to successfully reduce these costs is the dynamical low-rank approximation (DLRA). One key question when using DLRA methods is the construction of robust time integrators that preserve the invariances and associated conservation laws of the original problem. In this work, we demonstrate that the augmented basis update & Galerkin integrator (BUG) preserves solution invariances and the associated conservation laws when using a conservative truncation step and an appropriate time and space discretization. We present numerical comparisons to existing conservative integrators and discuss advantages and disadvantages. © 2023, The Authors. All rights reserved.

关键词： Integral equations

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Isometric Representations in Neural Networks Improve Robustness

arXiv

引用

arXiv 2022年

作者： Beshkov, Kosio Verhellen, Jonas Lepperød, Mikkel Elle Department of Biosciences University of Oslo Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Norway

Artificial and biological agents are unable to learn given completely random and unstructured data. The structure of data is encoded in the distance or similarity relationships between data points. In the context of neural networks, the neuronal activity within a layer forms a representation reflecting the transformation that the layer implements on its inputs. In order to utilize the structure in the data in a truthful manner, such representations should reflect the input distances and thus be continuous and isometric. Supporting this statement, recent findings in neuroscience propose that generalization and robustness are tied to neural representations being continuously differentiable. However, in machine learning, most algorithms lack robustness and are generally thought to rely on aspects of the data that differ from those that humans use, as is commonly seen in adversarial attacks. During cross-entropy classification, the metric and structural properties of network representations are usually broken both between and within classes. This side effect from training can lead to instabilities under perturbations near locations where such structure is not preserved. One of the standard solutions to obtain robustness is to train specifically by introducing perturbations in the training data. This leads to networks that are particularly robust to specific training perturbations but not necessarily to general perturbations. While adding ad hoc regularization terms to improve robustness has become common practice, to our knowledge, forcing representations to preserve the metric structure of the input data as a stabilising mechanism has not yet been introduced. In this work, we train neural networks to perform classification while simultaneously maintaining the metric structure within each class, leading to continuous and isometric within-class representations. We show that such network representations turn out to be a beneficial component for making accurate and robust

关键词： Network architecture

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Conservation properties of the augmented basis update & Galerkin integrator for kinetic problems

arXiv

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arXiv 2023年

作者： Einkemmer, Lukas Kusch, Jonas Schotthöfer, Steffen University of Innsbruck Numerical Analysis and Scientific Computing Innsbruck Austria Norwegian University of Life Sciences Scientific Computing Ås Norway Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

关键词： Kinetics

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Exploring Biologically Inspired Mechanisms of Adversarial Robustness

arXiv

引用

arXiv 2024年

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 Reserach 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 towards 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. © 2024, CC BY-NC-ND.

关键词： Adversarial machine learning

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

arXiv

引用

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

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DISCONTINUOUS GALERKIN METHODS FOR 3D–1D SYSTEMS

arXiv

引用

arXiv 2023年

作者： Masri, Rami Kuchta, Miroslav Riviere, Beatrice Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX77005 United States

We propose and analyze discontinuous Galerkin (dG) approximations to 3D-1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators. For the time dependent problem, a backward Euler dG formulation is also presented and analysed. Further, we propose a dG method for networks embedded in 3D domains, which is, up to jump terms, locally mass conservative on bifurcation points. numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our *** Codes 65N30, 65M60 Copyright © 2023, The Authors. All rights reserved.

关键词： Galerkin methods

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OPTIMAL CONTROL OF THE LANDAU-DE GENNES MODEL OF NEMATIC LIQUID CRYSTALS

arXiv

引用

arXiv 2023年

作者： Surowiec, Thomas M. Walker, Shawn W. Simula Research Laboratory Department of Numerical Analysis and Scientific Computing Kristian Augusts gate 23 Oslo0164 Norway Louisiana State University Baton RougeLA70803 United States

We present an analysis and numerical study of an optimal control problem for the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter Q = Q(x). Equilibrium LC states correspond to Q functions that (locally) minimize an LdG energy functional. Thus, we consider an L2-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semi-linear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external "force" controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where Q(x) = 0) in desired locations, which is desirable in *** Codes 49M25, 35K91, 65N30 © 2023, CC BY.

关键词： Finite element method

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analysis OF THE SIMPL METHOD FOR DENSITY-BASED TOPOLOGY OPTIMIZATION

arXiv

引用

arXiv 2024年

作者： Keith, Brendan Kim, Dohyun Lazarov, Boyan S. Surowiec, Thomas M. Division of Applied Mathematics Brown University ProvidenceRI02912 United States Lawrence Livermore National Laboratory LivermoreCA94550 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [31]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. We refer to the new algorithm as the SiMPL method, pronounced like "simple", which stands for Sigmoidal Mirror descent with a Projected Latent *** Codes 68Q25, 68R10, 68U05 Copyright © 2024, The Authors. All rights reserved.

关键词： Lagrange multipliers

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