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In recent years, a plethora of methods combining deep neural networks and partial differential equations have been developed. A widely known and popular example are physics-informed neural networks. They solve forward and inverse problems…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…
In many mechanistic medical, biological, physical and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs) can make simulations impractically slow. Biological models require the…
In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs)…
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…
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 introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part…
In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been…
The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains, are strongly affected by the amplitude of the Gaussian weight function employed to describe…
Motivated by the gap between theoretical optimal approximation rates of deep neural networks (DNNs) and the accuracy realized in practice, we seek to improve the training of DNNs. The adoption of an adaptive basis viewpoint of DNNs leads to…
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…
In the present paper we are concerned with a numerical algorithm for the approximation of the two-dimensional neural field equation with delay. We consider three numerical examples that have been analysed before by other authors and are…
This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for…
As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…
Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition,…
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
The objective of this paper is to investigate the ability of physics-informed neural networks to learn the magnetic field response as a function of design parameters in the context of a two-dimensional (2-D) magnetostatic problem. Our…
Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine…
Neural networks, a central tool in machine learning, have demonstrated remarkable, high fidelity performance on image recognition and classification tasks. These successes evince an ability to accurately represent high dimensional…