Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations

2020 ◽  
Vol 28 (5) ◽  
pp. 2002-2041
Author(s):  
Ameya D. Jagtap & George Em Karniadakis
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Domain Decomposition ◽  
Time Domain ◽  
Nonlinear Partial Differential Equations ◽  
Space Time ◽  
Partial Differential ◽  
Learning Framework

An acceleration scheme for deep learning-based BSDE solver using weak expansions

2020 ◽  
Vol 07 (02) ◽  
pp. 2050012
Author(s):  
Riu Naito ◽  
Toshihiro Yamada
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Backward Stochastic Differential Equation ◽  
Nonlinear Partial Differential Equations ◽  
High Dimensional ◽  
Parabolic Partial Differential Equations ◽  
Weak Approximation ◽  
Learning Method ◽  
Partial Differential

This paper gives an acceleration scheme for deep backward stochastic differential equation (BSDE) solver, a deep learning method for solving BSDEs introduced in Weinan et al. [Weinan, E, J Han and A Jentzen (2017). Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5(4), 349–380]. The solutions of nonlinear partial differential equations are quickly estimated using technique of weak approximation even if the dimension is high. In particular, the loss function and the relative error for the target solution become sufficiently small through a smaller number of iteration steps in the new deep BSDE solver.

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 The New Simulation of Quasiperiodic Wave, Periodic Wave, and Soliton Solutions of the KdV-mKdV Equation via a Deep Learning Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yong Zhang ◽  
Huanhe Dong ◽  
Jiuyun Sun ◽  
Zhen Wang ◽  
Yong Fang ◽  
...  
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Learning Method ◽  
Partial Differential

How to solve the numerical solution of nonlinear partial differential equations efficiently and conveniently has always been a difficult and meaningful problem. In this paper, the data-driven quasiperiodic wave, periodic wave, and soliton solutions of the KdV-mKdV equation are simulated by the multilayer physics-informed neural networks (PINNs) and compared with the exact solution obtained by the generalized Jacobi elliptic function method. Firstly, the different types of solitary wave solutions are used as initial data to train the PINNs. At the same time, the different PINNs are applied to learn the same initial data by selecting the different numbers of initial points sampled, residual collocation points sampled, network layers, and neurons per hidden layer, respectively. The result shows that the PINNs well reconstruct the dynamical behaviors of the quasiperiodic wave, periodic wave, and soliton solutions for the KdV-mKdV equation, which gives a good way to simulate the solutions of nonlinear partial differential equations via one deep learning method.

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.

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