Related papers: Fredholm Neural Networks
Regression with non-Euclidean responses -- e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions -- has become increasingly important in modern applications. In this paper, we propose deep…
In this paper, the authors propose the utilization of Fibonacci Neural Networks (FNN) for solving arbitrary order differential equations. The FNN architecture comprises input, middle, and output layers, with various degrees of Fibonacci…
Deep neural networks (DNNs) achieve remarkable performance on a wide range of tasks, yet their mathematical analysis remains fragmented: stability and generalization are typically studied in disparate frameworks and on a case-by-case basis.…
Machine learning algorithms have been successfully used to approximate nonlinear maps under weak assumptions on the structure and properties of the maps. We present deep neural networks using dense and convolutional layers to solve an…
Deep learning has delivered its powerfulness in many application domains, especially in image and speech recognition. As the backbone of deep learning, deep neural networks (DNNs) consist of multiple layers of various types with hundreds to…
Recently, machine learning methods have gained significant traction in scientific computing, particularly for solving Partial Differential Equations (PDEs). However, methods based on deep neural networks (DNNs) often lack convergence…
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…
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…
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…
Inverse problems for Partial Differential Equations (PDEs) are crucial in numerous applications such as geophysics, biomedical imaging, and material science, where unknown physical properties must be inferred from indirect measurements. In…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
Solving partial differential equations (PDEs) is an important yet challenging task in fluid mechanics. In this study, we embed an improved Fourier series into neural networks and propose a physics-informed Fourier basis neural network…
Data-driven machine learning approaches are being increasingly used to solve partial differential equations (PDEs). They have shown particularly striking successes when training an operator, which takes as input a PDE in some family, and…
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…
Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their…
The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the…
Semiparametric statistics play a pivotal role in a wide range of domains, including but not limited to missing data, causal inference, and transfer learning, to name a few. In many settings, semiparametric theory leads to (nearly)…
To help understand the underlying mechanisms of neural networks (NNs), several groups have, in recent years, studied the number of linear regions $\ell$ of piecewise linear functions generated by deep neural networks (DNN). In particular,…
We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a…
Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g.,…