Dual variational neural network decouples p-Laplace into linear Poisson and divergence-free minimization subproblems approximated by two NNs, with error analysis from vector inequalities and statistical learning theory, showing robust convergence for p near 1 and large p.
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For single- and two-index neural hypotheses in the deep Ritz method for the Schrödinger equation, gradient descent converges in O(log(1/ε)) iterations and the Ritz minimizer aligns with the source feature; a second feature emerges as regularization varies.
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Dual Variational Neural Network for the $p$-Laplace Problem
Dual variational neural network decouples p-Laplace into linear Poisson and divergence-free minimization subproblems approximated by two NNs, with error analysis from vector inequalities and statistical learning theory, showing robust convergence for p near 1 and large p.