scholarly journals Solving Partial Differential Equations Using Deep Learning and Physical Constraints

2020 ◽  
Vol 10 (17) ◽  
pp. 5917
Author(s):  
Yanan Guo ◽  
Xiaoqun Cao ◽  
Bainian Liu ◽  
Mei Gao
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Language Processing ◽  
Kdv Equation ◽  
Great Success ◽  
Partial Differential ◽  
Physical Constraints

The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task. Since many partial differential equations do not have analytical solutions, numerical methods are widely used to solve PDEs. Although numerical methods have been widely used with good performance, researchers are still searching for new methods for solving partial differential equations. In recent years, deep learning has achieved great success in many fields, such as image classification and natural language processing. Studies have shown that deep neural networks have powerful function-fitting capabilities and have great potential in the study of partial differential equations. In this paper, we introduce an improved Physics Informed Neural Network (PINN) for solving partial differential equations. PINN takes the physical information that is contained in partial differential equations as a regularization term, which improves the performance of neural networks. In this study, we use the method to study the wave equation, the KdV–Burgers equation, and the KdV equation. The experimental results show that PINN is effective in solving partial differential equations and deserves further research.

scholarly journals Deep Learning for Solving Partial Differential Equations

2020 ◽  
pp. 22-28
Author(s):  
А. А. Епифанов
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Deep Neural Networks ◽  
Learning Networks ◽  
Partial Differential ◽  
The Poisson Equation ◽  
The Galerkin Method

Глубокие нейронные сети стремительно развиваются в связи со значительным прогрессом в технологиях производительных вычислений. В данной работе рассматривается применение подходов, в основе которых лежит использование глубоких нейронных сетей, для решения дифференциальных уравнений в частных производных. Приводится пример численного решения уравнения Пуассона в двухмерной области методом Галеркина с глубокими нейронными сетями. Recently deep learning networks made huge progress due to the advances in highperformance computing technologies. This study covers a range of approaches to solving partial differential equations with deep learning. An example of solving the Poisson equation in a twodimensional domain using the Galerkin method with deep neural networks is presented.

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

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