Related papers: Parametric Complexity Bounds for Approximating PDE…
While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1)…
We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically…
In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods…
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
The aim of this work is to consider multiscale algorithms for solving PDEs with Galerkin methods on bounded domains. We provide results on convergence and condition numbers. We show how to handle PDEs with Dirichlet boundary conditions. We…
We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the…
We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this…
We investigate non-adaptive methods of deep ReLU neural network approximation in Bochner spaces $L_2({\mathbb U}^\infty, X, \mu)$ of functions on ${\mathbb U}^\infty$ taking values in a separable Hilbert space $X$, where ${\mathbb…
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower…
In this paper we consider PIDEs with gradient-independent Lipschitz continuous nonlinearities and prove that deep neural networks with ReLU activation function can approximate solutions of such semilinear PIDEs without curse of…
In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such…
We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we…
To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…
Recently a number of papers have suggested using neural-networks in order to approximate policy functions in DSGE models, while avoiding the curse of dimensionality, which for example arises when solving many HANK models, and while…
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…