Variational Physics-Informed Neural Networks for the $p(x)$-Laplacian Problem

Abstract

In this paper, we study a variational physics-informed neural network (VPINN) for solving the p(x)-Laplacian equation with Dirichlet boundary conditions. The proposed method builds a neural network that automatically satis es the boundary values and minimizes the energy of the problem. The loss function is computed using Monte Carlo sampling, and derivatives are obtained with automatic di erentiation. To train the network, we use two steps: rst the Adam optimizer, then the L-BFGS method for faster convergence. We test the approach on several examples in one and two dimensions, where the exponent $p(x)$ changes smoothly or has jumps. The results show that the VPINN gives accurate and stable solutions, even when the coe cients vary strongly in space.