Related papers: Parametric Complexity Bounds for Approximating PDE…

Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks…

It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent…

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a…

In recent years residual neural networks (ResNets) as introduced by [He, K., Zhang, X., Ren, S., and Sun, J., Proceedings of the IEEE conference on computer vision and pattern recognition (2016), 770-778] have become very popular in a large…

Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…

Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…

Whilst the Universal Approximation Theorem guarantees the existence of approximations to Sobolev functions -- the natural function spaces for PDEs -- by Neural Networks (NNs) of sufficient size, low-regularity solutions may lead to poor…

Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…

In this paper, we perform the convergence analysis of unsupervised Legendre--Galerkin neural networks (ULGNet), a deep-learning-based numerical method for solving partial differential equations (PDEs). Unlike existing deep learning-based…

We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in $H^1(\Omega)$ for weighted analytic function classes in certain polytopal domains $\Omega$, in space dimension $d=2,3$. Functions in these classes are locally…

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…

In this article, we investigate the existence of a deep neural network (DNN) capable of approximating solutions to partial integro-differential equations while circumventing the curse of dimensionality. Using the Feynman-Kac theorem, we…

In this paper we prove that rectified deep neural networks do not suffer from the curse of dimensionality when approximating McKean--Vlasov SDEs in the sense that the number of parameters in the deep neural networks only grows polynomially…

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…

We explore conditions for when the gradient of a deep declarative node can be approximated by ignoring constraint terms and still result in a descent direction for the global loss function. This has important practical application when…

This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any H\"{o}lder smooth…

Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It…

The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural…

In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove the proposed multiscale method achieves…

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…