PINN-DG replaces pointwise derivative losses with finite element interpolation plus discontinuous Galerkin consistency and penalty terms, and proves convergence of the discrete minimizers.
Error estimates of residual minimization using neural networks for linear PDEs
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abstract
We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates for both continuous and discrete formulations of residual minimization in strong and weak forms. The formulations cover recently developed physics-informed neural networks based on strong and variational formulations.
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PINN-DG: Residual neural network methods trained with Finite Elements
PINN-DG replaces pointwise derivative losses with finite element interpolation plus discontinuous Galerkin consistency and penalty terms, and proves convergence of the discrete minimizers.