Related papers: Neural network augmented inverse problems for PDEs

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

Inverse problems exist in a wide variety of physical domains from aerospace engineering to medical imaging. The goal is to infer the underlying state from a set of observations. When the forward model that produced the observations is…

This paper proposes a deep learning architecture that attains statistically significant improvements over traditional algorithms in Poisson image denoising espically when the noise is strong. Poisson noise commonly occurs in low-light and…

Artificial and biological agents cannon learn given completely random and unstructured data. The structure of data is encoded in the metric relationships between data points. In the context of neural networks, neuronal activity within a…

We suggest a general approach to quantification of different forms of aleatoric uncertainty in regression tasks performed by artificial neural networks. It is based on the simultaneous training of two neural networks with a joint loss…

Image denoising methods must effectively model, implicitly or explicitly, the vast diversity of patterns and textures that occur in natural images. This is challenging, even for modern methods that leverage deep neural networks trained to…

Solving inverse problems is a fundamental component of science, engineering and mathematics. With the advent of deep learning, deep neural networks have significant potential to outperform existing state-of-the-art, model-based methods for…

Adversarial images are designed to mislead deep neural networks (DNNs), attracting great attention in recent years. Although several defense strategies achieved encouraging robustness against adversarial samples, most of them fail to…

Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…

We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…

In this Master Thesis, we study the approximation capabilities of Neural Networks in the context of numerical resolution of elliptic PDEs and Approximation Theory. First of all, in Chapter 1, we introduce the mathematical definition of…

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

Networks are powerful instruments to study complex phenomena, but they become hard to analyze in data that contain noise. Network backbones provide a tool to extract the latent structure from noisy networks by pruning non-salient edges. We…

In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most…

Model compression techniques allow to significantly reduce the computational cost associated with data processing by deep neural networks with only a minor decrease in average accuracy. Simultaneously, reducing the model size may have a…

In inverse problems, it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropriately represented…

Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the…

The Poisson equation is critical to get a self-consistent solution in plasma fluid simulations used for Hall effect thrusters and streamer discharges, since the Poisson solution appears as a source term of the unsteady nonlinear flow…

We explore artificial neural networks as a tool for the reconstruction of spectral functions from imaginary time Green's functions, a classic ill-conditioned inverse problem. Our ansatz is based on a supervised learning framework in which…

We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy…