Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. In the Deep-Ritz method, proposed by professor E. Weinan, how to optimize the neural network to make it more accurate has become a problem worthy of attention. In our work, we have conducted a comparative study on the network structure and different optimization methods. In terms of network structure, primarily, we introduced the RBF activation function, combined it with the ResNet network, and proposed the Combined-Ritz network (CRM). Comparing it with DRM and RRM (the network simply based on RBF function), We can see that in the case of low-dimensionality, DRM converges slowly and has many parameters, but the accuracy is higher. RRM converges fast, has fewer parameters, and has a lower accuracy. CRM combines the advantages of the two with fewer parameters and higher accuracy. In addition, in the two-dimensional situation, we proposed the CNN network architecture to solve the partial differential equation problem, and achieved good success.
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