Neural network method for solving differential equation

Abstract

The invention discloses a neural network method for solving a differential equation, and the method comprises the following steps: (1), carrying out the sampling of data of each dimension in a definition domain, and obtaining a trained overall data set; (2) automatically adjusting the training data in each round to obtain the training data of each round; (3) constructing a physical information neural network according to the condition of the differential equation, and solving the differential equation; regression calculation is carried out on parameters of the network for multiple times in a neural network solving differential equation, and each regression calculation is called as a round. According to the technical scheme, the distribution of the obtained data points in the definition domain range is more characteristic in a hierarchical data sampling mode through the normal distribution cumulative density function, and sampling across definition domain boundary points not only facilitates training of the data points near the network boundary, but also facilitates training of the data points near the network boundary. The method also has better solving precision in conditions such as a periodic boundary, and effectively improves the overall training precision of the network.

Description

technical field [0001] The invention relates to the technical field of solving partial differential equations, in particular to a neural network method for solving differential equations. Background technique [0002] In many fields such as physics, engineering, and medical treatment, differential equations are often used to model and analyze specific problems. According to the diversity and complexity of the problem, the dimension of the differential equation is gradually deepened, and the difficulty of solving it is also gradually deepened. The analytical solutions of such differential equations are difficult to calculate, and numerical methods such as finite difference method and finite element method are usually used to solve the numerical solutions of differential equations. However, these methods are highly targeted, and the construction is complex, making it inconvenient to transplant. [0003] With the proof of the universal approximation of the neural network stru...

AI Technical Summary

Problems solved by technology

[0006] For some data points that have not been trained, the neural network shows that the network is difficult to train, and there is a large error between the training effect and the real numerical solution.

Especially for the boundary condition problems of differential equations, the data points at the boundary have a low probability of being obtained, so there are often large errors

Differential equation solving methods based on neural networks, such as PINNS, have problems such as the network cannot be optimized, the loss function is difficult to converge, and the gradient disappears in the solution process.

Causes the network to be unable to find a solution to the differential equation

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