In this paper,hybrid physics-informed neural networks(PINNs)are pro-posed for solving partial differential equations(PDEs).In this approach,we introduce a difference scheme based on local mesh to construct the physical residuals as part of the loss function.The obvious advantage is that the hybrid PINNs are not completely dependent on automatic differentiation techniques and are more sensitive to gradient changes in the solution.In addition,since the PINNs are continuous mappings,local meshes at arbitrary points can be built.Therefore,all local meshes are independent and the hybrid PINNs are not limited by dimension.Finally,the performance of the hybrid PINNs is verified by numerical experiments,and the effects of the order of differential schemes and the size of local mesh on accuracy are discussed.The results show that the generalization ability of the hybrid PINNs is more significant better than that of the PINNs.
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