We propose to enhance the training of physics-informed neural networks (PINNs). To this aim, we introduce nonlinear additive and multiplicative preconditioning strategies for the widely used L-BFGS optimizer. The nonlinear preconditioners are constructed by utilizing the Schwarz domain-decomposition framework, where the parameters of the network are decomposed in a layer-wise manner. Through a series of numerical experiments, we demonstrate that both, additive and multiplicative, preconditioners significantly improve the convergence of the standard L- BFGS optimizer, while providing more accurate solutions of underlying partial differential equations. Moreover, the additive preconditioner is inherently parallel, thus giving rise to a novel approach to model parallelism.