Auxiliary Finite-Difference Residual-Gradient Regularization for PINNs

arXiv:2604.14472v1 Announce Type: cross Abstract: Physics-informed neural networks (PINNs) are often selected by a single scalar loss even when the quantity of interest is more specific. We study a hybrid design in which the governing PDE residual remains automatic-differentiation (AD) based, while finite differences (FD) appear only in a weak auxiliary term that penalizes gradients of the sampled residual field. The FD term regularizes the residual field without replacing the PDE residual itsel