Robust training and rigorous error analysis of physics-informed neural networks for the $p$-Laplace equation

What the math gives to ML

The paper's transferable contribution is to replace pointwise PDE residual penalties with residuals measured in the dual space W^{-1,p'}, which is better aligned with weak solutions and limited regularity. The dual norm turns residual training into an adversarial weak-form test: the network is penalized according to the worst normalized test function rather than the largest pointwise derivative error. Its boundary treatment also suggests matching traces in the fractional space W^{1-1/p,p}(partial Omega), avoiding an unnecessarily strong pointwise boundary penalty. A practical first transfer is a PINN loss with alternating neural test-function maximization and a Monte Carlo approximation of the fractional boundary seminorm.