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Turing dynamics mainly focuses on non-homogeneous self-adaptive spatial patterns of reaction diffusion systems in a continuous space. If defining the diffusion environment as network structure, then network patterns are available under Turing instability conditions. In allusion to parameter estimation for network Turing-instable systems, this paper proposes a new recognition method using artificial neural networks. When diffusion occurs on quadrilateral lattice networks, a deep convolutional neural network (CNN) is built to invert the unknown parameters. The results on training set and validation set indicate that the CNN model exhibits great robustness without overfitting and gradient explosion. Meanwhile, the prediction performance on two test sets is excellent, with average relative errors of the estimators ending up at 0.68% and 1.04%, respectively. When diffusion occurs on non-regular networks, a spatial domain graph convolutional network (GCN) is established, which is slightly less robust than the CNN. The average relative error of the estimators on both test sets ends up between 1.1% and 2.8%, which is still in the ideal range. • The parameter estimation method for network Turing systems is investigated. • A 14-layer deep neural network is designed for systems on lattice networks. • A 6-layer graph neural network is designed for systems on irregular networks. [ABSTRACT FROM AUTHOR]