Related papers: Computational characteristics of feedforward neura…
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…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
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…
Fractional differential equations are powerful mathematical descriptors for intricate physical phenomena in a compact form. However, compared to integer ordinary or partial differential equations, solving fractional differential equations…
A method to increase the precision of feedforward networks is proposed. It requires a prior knowledge of a target function derivatives of several orders and uses this information in gradient based training. Forward pass calculates not only…
Differential equations are used in a wide variety of disciplines, describing the complex behavior of the physical world. Analytic solutions to these equations are often difficult to solve for, limiting our current ability to solve complex…
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,…
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,…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations,…
We motivate the use of neural networks for the construction of numerical solutions to differential equations. We prove that there exists a feed-forward neural network that can arbitrarily minimise an objective function that is zero at the…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
In this paper, feedforward neural networks are presented that have nonlinear weight functions based on look--up tables, that are specially smoothed in a regularization called the diffusion. The idea of such a type of networks is based on…
Neural Ordinary Differential Equations (Neural ODEs) represent a significant breakthrough in deep learning, promising to bridge the gap between machine learning and the rich theoretical frameworks developed in various mathematical fields…
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…
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…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
Deep learning has become the state-of-art tool in many applications, but the evaluation and training of deep models can be time-consuming and computationally expensive. The conditional computation approach has been proposed to tackle this…
We revisit the original approach of using deep learning and neural networks to solve differential equations by incorporating the knowledge of the equation. This is done by adding a dedicated term to the loss function during the optimization…
Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance one may include…