Abstract
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.
Highlights
Partial differential equations (PDEs) are important tools for the study of all kinds of natural phenomena and they are widely used to explain various physical laws [1,2,3]
This paper introduces a method for solving partial differential equations by neural networks that fuse physical information
The physical laws that are contained in the partial differential equations are introduced into the neural networks as a regularization
Summary
Partial differential equations (PDEs) are important tools for the study of all kinds of natural phenomena and they are widely used to explain various physical laws [1,2,3]. Numerical experiments show that the method can achieve reasonable physical results using a smaller number of neurons, reducing the memory requirements This method still faces problems, such as slow training speed and overfitting, which may be solved by dropout or regularization. Huang [28] combines deep neural networks and the Wiener-Hopf method to study some wave problems This combinational research strategy has achieved excellent experimental results in solving two particular problems. Jagtap et al [37] introduce adaptive activation functions into deep and physics-informed neural networks (PINNs) to better approximate complex functions and the solutions of partial differential equations. The paper is structured, as follows: Section 2 introduces the proposed algorithm for solving partial differential equations based on neural networks and physical knowledge constraints.
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