Related papers: Exponential Convergence of Deep Operator Networks …
In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography (EIT), where the goal is to recover the conductivity in a medium from boundary current-to-voltage measurements, i.e., the…
It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from…
Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with…
This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential…
We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated…
This paper proposes a Kolmogorov high order deep neural network (K-HOrderDNN) for solving high-dimensional partial differential equations (PDEs), which improves the high order deep neural networks (HOrderDNNs). HOrderDNNs have been…
Deep operator networks (DeepONets) are trained to predict the linear amplification of instability waves in high-speed boundary layers and to perform data assimilation. In contrast to traditional networks that approximate functions,…
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…
Operator learning problems arise in many key areas of scientific computing where Partial Differential Equations (PDEs) are used to model physical systems. In such scenarios, the operators map between Banach or Hilbert spaces. In this work,…
In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to…
We consider the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain $D$ of $\mathbb{R}^d$, with $d \geq 2$. In this paper, we perform a stochastic gradient descent algorithm that approximates the…
Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seed numerical solutions for a class of partial differential equations instead of a single partial differential equation,…
This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on…
Deep learning method is of great importance in solving partial differential equations. In this paper, inspired by the failure-informed idea proposed by Gao et.al. (SIAM Journal on Scientific Computing 45(4)(2023)) and as an improvement, a…
Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of…
In this paper, we investigate 1D elliptic equations $-\nabla\cdot (a\nabla u)=f$ with rough diffusion coefficients $a$ that satisfy $0<a_{\min}\le a\le a_{\max}<\infty$ and $f\in L_2(\Omega)$. To achieve an accurate and robust numerical…
End-to-end deep neural networks (DNNs) have become the state-of-the-art (SOTA) for solving inverse problems. Despite their outstanding performance, during deployment, such networks are sensitive to minor variations in the testing pipeline…
Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning…
Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep…