To enhance the prediction accuracy of total organic carbon(TOC), and according to time-varying, singularity feature of logging curve, the ridgelet transform function is used as the activation function for process neuron and a continuous ridgelet process neural network is proposed. Firstly, the gradient descent method based on orthogonal basis expansion is proposed. Then, in order to improve the training convergence ability, a quantum-inspired cuckoo search algorithm is proposed and applied to model training, in which, the individual's Lévy flight follows the latitude on the Bloch sphere. Finally, the trained ridgelet process neural network is applied to shale TOC prediction. Some logging curves which have sensitive response to TOC are selected as the model feature inputs by the correlation analysis. Through the comparison with other process neural networks, the experimental result shows that the TOC prediction accuracy increases about 7 percent.