In surrogate modelling, a simple functional approximation of a complex system model is always constructed to reduce the computational expense, and the selection of a suitable surrogate model and a sampling method are key to obtaining a surrogate model for a complex system. To construct an appropriate surrogate model, three methods of adaptive surrogate modelling that use artificial neural networks (ANN) are developed by incorporating a new mechanism for automatically determining the number of hidden nodes and/or a new prediction error-based mixed adaptive sampling method. In the automatic determination, the number of hidden nodes can adaptively change according to the effective rate of parameters in the ANN during the adaptive surrogate modelling process. As a result, an improper number of hidden nodes determined by the empirical method can be avoided. The prediction error-based mixed adaptive sampling method is capable of finding the strong nonlinear behaviour of the underlying system, which is easily missed by the traditional prediction variance-based sampling method. The three methods and the previous method for adaptive surrogate modelling that use ANN are tested and compared in terms of replicating the behaviours of three types of challenge functions to determine the efficacy of the developed methods. Furthermore, these methods are used in an engineering problem of surrogate modelling for a cracking reaction simulator to validate the efficacy of the developed methods.