approximation

Finite element approximation of parameter dependent eigenvalue problems

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B9 L2 R2322

Eigenvalue problems arising from partial differential equations are used to model several applications in science and engineering, ranging from vibrations of structures, industrial microwaves, photonic crystals, and waveguides, to particle accelerators.

Efficient Computation Through Tuned Approximation

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B9 L2 R2322 H1

Numerical software is being reinvented to provide opportunities to tune dynamically the accuracy of computation to the requirements of the application, resulting in savings of memory, time, and energy. Floating point computation in science and engineering has a history of “oversolving” relative to expectations for many models. So often are real datatypes defaulted to double precision that GPUs did not gain wide acceptance until they provided in hardware operations not required in their original domain of graphics. However, computational science is now reverting to employ lower precision arithmetic where possible. Many matrix operations considered at a blockwise level allow for lower precision and many blocks can be approximated with low rank near equivalents.

Deep Neural Network Approximation via Function Compositions

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KAUST

Abstract Deep neural networks have brought transformative changes in areas such as image recognition and healthcare, yet their underlying theoretical principles remain somewhat elusive. The main emphasis of this presentation is to explore the theoretical dimensions of deep neural networks, with a special focus on evaluating their approximation capabilities. We begin by assessing the approximation errors in ReLU networks when they attempt to approximate various target functions, including (Lipschitz) continuous functions, polynomials, and smooth functions. Utilizing the concept of Vapnik

Approximation and Generalization Errors in Deep Neural Networks for Sobolev Spaces measured by Sobolev Norms

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KAUST

Abstract In this presentation, we initially discuss the approximation capabilities of deep neural networks (DNNs) with ReLU and the square of ReLU as activation functions for Sobolev functions measured by the Sobolev norms \(W^{m,p}\) where \(m \ge 1\). Subsequently, we consider how to address the issue of the curse of dimensionality for DNNs’ approximation. Finally, we analyze the generalization errors associated with DNNs using such Sobolev loss functions. Additionally, we provide recommendations on when to opt for deeper NNs versus wider NNs, considering factors such as the number of sample

WAN Approximation of PDEs: Best Approximation, Stabilization and Boundary Conditions

Prof. Silvia Bertoluzza

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B2 L5 R5209

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework.