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On choosing interpolation domains and functions in topology optimization of multi-scale structures: a comparative study

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Structural and Multidisciplinary Optimization 2025年第6期68卷 1-20页

作者： Ali, Ahmed Mohamed Jubartalla Altassan, Abdulmajeed Cherrière, Théodore Gangl, Peter Gföhler, Margit Kapl, Mario ADMiRE Research Center Carinthia University of Applied Sciences Villach Austria Research Unit of Biomechanics and Rehabilitation Engineering TU Wien Vienna Austria George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta USA Johann Radon Institute for Computational and Applied Mathematics (RICAM) Linz Austria Department of Engineering & IT Carinthia University of Applied Sciences Villach Austria

In this work, we evaluate the behavior of several material interpolation schemes in conducting topology optimization of composite structures made of 28 orthotropic materials with similar costs and densities. The materials are categorized into three families according to their geometrical features and arranged in a collection of interpolation domains, including 2D and 3D domains, as well as orthogonal and non-orthogonal ones. The concept of Coons patches is used to extend three interpolation functions, namely polynomial fitting, ordered Solid Isotropic Material with Penalization (ordered SIMP), and smooth single-variable-based interpolation, from 1D to 2D and 3D. Several material interpolation schemes are built using different combinations of interpolation domains and interpolation functions for a representative comparison. Numerical experiments are conducted using a two-scale topology optimization procedure to solve a benchmark problem given by the Messerschmitt-Bölkow-Blohm (MBB) beam problem. The obtained results show that, when taking the simple case of compliance minimization, as well as candidate lattices of same densities, the selection of the interpolation domain and of the interpolation function does not have a significant effect on the optimization process. Accordingly, we conclude that taking the simplest combination of a 2D square domain together with a piecewise linear function is sufficient to get the optimized design, if the candidate lattices are well categorized and sorted.

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ON THE SPECIALIZATION OF NEURAL MODULES

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arXiv 2024年

作者： Jarvis, Devon Klein, Richard Rosman, Benjamin Saxe, Andrew M. School of Computer Science and Applied Mathematics University of the Witwatersrand South Africa Gatsby Computational Neuroscience Unit Sainsbury Wellcome Centre UCL United Kingdom CIFAR Azrieli Global Scholar CIFAR Canada

A number of machine learning models have been proposed with the goal of achieving systematic generalization: the ability to reason about new situations by combining aspects of previous experiences. These models leverage compositional architectures which aim to learn specialized modules dedicated to structures in a task that can be composed to solve novel problems with similar structures. While the compositionality of these architectures is guaranteed by design, the modules specializing is not. Here we theoretically study the ability of network modules to specialize to useful structures in a dataset and achieve systematic generalization. To this end we introduce a minimal space of datasets motivated by practical systematic generalization benchmarks. From this space of datasets we present a mathematical definition of systematicity and study the learning dynamics of linear neural modules when solving components of the task. Our results shed light on the difficulty of module specialization, what is required for modules to successfully specialize, and the necessity of modular architectures to achieve systematicity. Finally, we confirm that the theoretical results in our tractable setting generalize to more complex datasets and non-linear architectures. Copyright © 2024, The Authors. All rights reserved.

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OPTIMAL RUNGE APPROXIMATION FOR NONLOCAL WAVE EQUATIONS AND UNIQUE DETERMINATION OF POLYHOMOGENEOUS NONLINEARITIES

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arXiv 2024年

作者： Lin, Yi-Hsuan Tyni, Teemu Zimmermann, Philipp Department of Applied Mathematics National Yang Ming Chiao Tung University Hsinchu Taiwan Fakultät für Mathematik University of Duisburg-Essen Essen Germany Research Unit of Applied and Computational Mathematics University of Oulu Finland Departament de Matemàtiques i Informàtica Universitat de Barcelona Barcelona Spain

The main purpose of this article is to establish the Runge-type approximation in L2(0, T;(eq)s(Ω)) for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calderón problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calderón problem of nonlocal wave equations with polyhomogeneous *** Codes Primary 35R30, Secondary 26A33, 42B37 Copyright © 2024, The Authors. All rights reserved.

关键词： Wave equations

Overcoming Saturation in Density Ratio Estimation by Iterated Regularization

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arXiv 2024年

作者： Gruber, Lukas Holzleitner, Markus Lehner, Johannes Hochreiter, Sepp Zellinger, Werner ELLIS Unit Linz and LIT AI Lab Institute for Machine Learning Johannes Kepler University Linz Austria MaLGa Center Department of Mathematics University of Genoa Italy NXAI GmbH Linz Austria Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Austria

Estimating the ratio of two probability densities from finitely many samples, is a central task in machine learning and statistics. In this work, we show that a large class of kernel methods for density ratio estimation suffers from error saturation, which prevents algorithms from achieving fast error convergence rates on highly regular learning problems. To resolve saturation, we introduce iterated regularization in density ratio estimation to achieve fast error rates. Our methods outperform its non-iteratively regularized versions on benchmarks for density ratio estimation as well as on large-scale evaluations for importance-weighted ensembling of deep unsupervised domain adaptation models. Copyright © 2024, The Authors. All rights reserved.

关键词： Machine learning

Overcoming saturation in density ratio estimation by iterated regularization 24

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Overcoming saturation in density ratio estimation by iterate...

Proceedings of the 41st International Conference on Machine Learning

作者： Lukas Gruber Markus Holzleitner Johannes Lehner Sepp Hochreiter Werner Zellinger ELLIS Unit Linz and LIT AI Lab Institute for Machine Learning Johannes Kepler University Linz Austria MaLGa Center Department of Mathematics University of Genoa ELLIS Unit Linz and LIT AI Lab Institute for Machine Learning Johannes Kepler University Linz Austria and NXAI GmbH Linz Austria Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences

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MAKE HASTE SLOWLY: A THEORY OF EMERGENT STRUCTURED MIXED SELECTIVITY IN FEATURE LEARNING RELU NETWORKS

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arXiv 2025年

作者： Jarvis, Devon Klein, Richard Rosman, Benjamin Saxe, Andrew M. School of Computer Science and Applied Mathematics University of the Witwatersrand South Africa Machine Intelligence and Neural Discovery Institute University of the Witwatersrand South Africa Gatsby Computational Neuroscience Unit Sainsbury Wellcome Centre UCL United Kingdom CIFAR Azrieli Global Scholar CIFAR Canada

In spite of finite dimension ReLU neural networks being a consistent factor behind recent deep learning successes, a theory of feature learning in these models remains elusive. Currently, insightful theories still rely on assumptions including the linearity of the network computations, unstructured input data and architectural constraints such as infinite width or a single hidden layer. To begin to address this gap we establish an equivalence between ReLU networks and Gated Deep Linear Networks, and use their greater tractability to derive dynamics of learning. We then consider multiple variants of a core task reminiscent of multi-task learning or contextual control which requires both feature learning and nonlinearity. We make explicit that, for these tasks, the ReLU networks possess an inductive bias towards latent representations which are not strictly modular or disentangled but are still highly structured and reusable between contexts. This effect is amplified with the addition of more contexts and hidden layers. Thus, we take a step towards a theory of feature learning in finite ReLU networks and shed light on how structured mixed-selective latent representations can emerge due to a bias for node-reuse and learning speed. © 2025, CC BY.

关键词： Multi-task learning

Coprime networks of the composite numbers: pseudo-randomness and synchronizability

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arXiv 2024年

作者： Miraj, Md Rahil Ghosh, Dibakar Hens, Chittaranjan Theoretical Statistics and Mathematics Unit Indian Statistical Institute 8th Mile Mysore Rd RVCE Post Karnataka Bengaluru560059 India Physics and Applied Mathematics Unit Indian Statistical Institute 203 B.T. Road West Bengal Kolkata700108 India Center for Computational Natural Science and Bioinformatics International Institute of Informational Technology Gachibowli Telengana Hyderabad500032 India

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index n ≥ 49. To investigate how the nodes are connected, we analytically describe that the link density saturates to 6/π2, whereas the average degree increases linearly with slope 6/π2 with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for 49 ≤ n ≤ 288 and it is at most 2 for n ≥ 289. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of r-length labeled cycles, which indicates the existence of a cycle of length at most O(log n). Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős–Rényi random network and Barabási-Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator-prey networks which can easily avoid the global synchronization. © 2024, CC BY.

关键词： Eigenvalues and eigenfunctions

A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on experimental data

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arXiv 2023年

作者： Tyni, Teemu Stinchcombe, Adam R. Alexakis, Spyros Unit of Applied and Computational Mathematics Faculty of Information Technology and Electrical Engineering University of Oulu Pentti Kaiteran Katu 1 Oulu90014 Finland Department of Mathematics University of Toronto 40 St. George Street TorontoONM5S 2E4 Canada

We develop a boundary integral equation-based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, and currents are injected through the electrodes and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single layer potentials;the problem is thus re-cast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This forward solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography (EIT) which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems *** Codes 65N21, 35R30 Copyright © 2023, The Authors. All rights reserved.

关键词： Electrodes

A THEORY OF INITIALISATION’S IMPACT ON SPECIALISATION

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arXiv 2025年

作者： Jarvis, Devon Lee, Sebastian Dominé, Clémentine Carla Juliette Saxe, Andrew M. Mannelli, Stefano Sarao School of Computer Science and Applied Mathematics University of the Witwatersrand South Africa Center for Computational Neuroscience Flatiron Institute Simons Foundation Gatsby Computational Neuroscience Unit Sainsbury Wellcome Centre UCL United Kingdom Data Science and AI Computer Science and Engineering Chalmers University of Technology University of Gothenburg Sweden Machine Intelligence and Neural Discovery Institute University of the Witwatersrand South Africa CIFAR Azrieli Global Scholar CIFAR

Prior work has demonstrated a consistent tendency in neural networks engaged in continual learning tasks, wherein intermediate task similarity results in the highest levels of catastrophic interference. This phenomenon is attributed to the network’s tendency to reuse learned features across tasks. However, this explanation heavily relies on the premise that neuron specialisation occurs, i.e. the emergence of localised representations. Our investigation challenges the validity of this assumption. Using theoretical frameworks for the analysis of neural networks, we show a strong dependence of specialisation on the initial condition. More precisely, we show that weight imbalance and high weight entropy can favour specialised solutions. We then apply these insights in the context of continual learning, first showing the emergence of a monotonic relation between task-similarity and forgetting in non-specialised networks. Finally, we show that specialization by weight imbalance is beneficial on the commonly employed elastic weight consolidation regularisation technique. Copyright © 2025, The Authors. All rights reserved.

关键词： Neural networks