Auxiliary Finite-Difference Residual-Gradient Regularization for PINNs

ArXi:2604.14472v1 Announce Type: cross Physics-informed neural networks (PINNs) are often selected by a single scalar loss even when the quantity of interest is 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 itself. We examine this idea in two stages.