Related papers: The Neural Network Approach to Inverse Problems in…
Finding model parameters from data is an essential task in science and engineering, from weather and climate forecasts to plasma control. Previous works have employed neural networks to greatly accelerate finding solutions to inverse…
We propose a unified framework for delay differential equations (DDEs) based on deep neural networks (DNNs) - the neural delay differential equations (NDDEs), aimed at solving the forward and inverse problems of delay differential…
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…
This paper studies deep neural networks for solving extremely large linear systems arising from highdimensional problems. Because of the curse of dimensionality, it is expensive to store both the solution and right-hand side vector in such…
Many phenomena in physics, including light, water waves, and sound, are described by wave equations. Given their coefficients, wave equations can be solved to high accuracy, but the presence of the wavelength scale often leads to large…
Inverse problems arise in a variety of imaging applications including computed tomography, non-destructive testing, and remote sensing. The characteristic features of inverse problems are the non-uniqueness and instability of their…
This review provides an introduction to - and overview of - the current state of the art in neural-network based regularization methods for inverse problems in imaging. It aims to introduce readers with a solid knowledge in applied…
Reconstructing spectral functions from Euclidean Green's functions is an important inverse problem in physics. The prior knowledge for specific physical systems routinely offers essential regularization schemes for solving the ill-posed…
While deep learning methods have achieved state-of-the-art performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach,…
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…
Neural network-based methods for solving differential equations have been gaining traction. They work by improving the differential equation residuals of a neural network on a sample of points in each iteration. However, most of them employ…
In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data,…
We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to…
We propose a general framework for solving forward and inverse problems constrained by partial differential equations, where we interpolate neural networks onto finite element spaces to represent the (partial) unknowns. The framework…
Inverse problems exist in many domains such as phase imaging, image processing, and computer vision. These problems are often solved with application-specific algorithms, even though their nature remains the same: mapping input image(s) to…
The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…
In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
A novel artificial neural network method is proposed for solving Cauchy inverse problems. It allows multiple hidden layers with arbitrary width and depth, which theoretically yields better approximations to the inverse problems. In this…