Radial Laplace modulation factor acceleration for solving seepage equations using asymptotic solution networks

Solving partial differential equations (PDEs) poses significant challenges, particularly for nonlinear PDEs with source-sink terms or drastic gradient changes. Traditional numerical methods often encounter high computational costs and difficulties in handling complex geometries. While physics-informed neural networks offer a promising alternative, they struggle with slow convergence and instability in intricate scenarios. To overcome these limitations, we introduce radial Laplace asymptotic solution networks (RL-ASNets), which incorporate a radial Laplace modulation factor into the network architecture. RL-ASNets consist of two main components: an asymptotic network and a correction network. The asymptotic network derives preliminary asymptotic solutions, providing initial approximations. Meanwhile, the correction network refines these solutions by addressing any errors in the asymptotic outputs. The radial Laplace modulation factor assigns differentiated weights to various spatial positions, emphasizing solutions in areas with significant gradient changes. This mechanism enhances both the convergence speed and the accuracy of simulations. Consequently, RL-ASNets better capture and simulate the complex solution structures in PDEs, thereby enhancing the ability of the model to simulate complex physical phenomena. The experimental results demonstrate that the performance of RL-ASNets significantly surpasses that of the comparative models, achieving a convergence speed over 13 times faster, and the solution error deviates from the numerical solution by only approximately 10−3.