Dual Variational Neural Network for the $p$-Laplace Problem
For researchers in numerical PDEs, this provides a stable deep learning method for a notoriously difficult nonlinear operator, though it is an incremental application of existing neural network techniques.
The paper introduces a dual variational neural network for solving $p$-Laplace problems, achieving robust convergence in extreme regimes $p \ o 1^{+}$ and $p \\gg 1$, as demonstrated by numerical experiments.
The reliable and accurate numerical approximation of the $p$-Laplacian is particularly challenging in the extreme regimes $p \to 1^{+}$ and $p \gg 1$, where the operator becomes either highly singular or strongly degenerate, often causing severe instability in standard numerical methods. To address these difficulties, we propose a novel deep learning based framework, termed the dual variational neural network, for $p$-Laplace problems. The approach is based on a mixed formulation and an $L^q$-based Helmholtz decomposition, which decouples the original problem into two convex subproblems: a linear Poisson problem for the irrotational component and an unconstrained minimization problem over divergence-free fields for the solenoidal component. Following the decomposition, we employ two neural networks using a gradient--curl representation to approximate the flux, and further establish an error analysis of the neural approximation. The analysis relies on fundamental vector inequalities together with tools from statistical learning theory. Numerical experiments demonstrate robust convergence of the proposed method in challenging settings, including the extreme cases $p \to 1^{+}$ and $p \gg 1$, as well as the $p(x)$-Laplace equation.