Physics-Informed Neural Networks (PINNs) have demonstrated significant potential in solving partial differential equations (PDEs) and modeling complex physical systems. However, when dealing with multi-scale, multi-domain scenarios and multi-physics coupled systems, PINNs face challenges such as low training efficiency and optimization instability. Based on existing PINN methods, a conservation-based Cross-Twin Network (CTN) approach is proposed for solving wave equations. By introducing interactive information-sharing and constraint mechanisms, the proposed method significantly improves the convergence speed, prediction accuracy, and training stability in multi-domain and multi-scale scenarios. Experimental results show that, compared with traditional methods, the Cross-Twin Network achieves superior performance in solving nonlinear higher-order wave PDEs and equation systems. This study provides new insights for the research and application of PINNs.

In this research, the anisotropic diffusion equation is solved based on physics-informed neural network (PINN) . Firstly, a vertical parallel sampling method is proposed to collect data points in different directions of the material coordinate system. The experimental results show that the method can accurately capture the anisotropic variation characteristics in the problem and improve the prediction accuracy of the neural network compared with the traditional random sampling method. Secondly, the weighted integral discretization method is proposed, which is different from the ordinary PINN method in that it changes the calculation of the data point error to the calculation of the integral-type error function, which is discretized by embedding the weight function and using the Gaussian integral numerical method. The experimental results show that the accuracy of the predicted results using this method has a significant improvement, and the convergence stability is stronger and the convergence speed is faster.