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Predicting the oil–gas-bearing distribution of unconventional reservoirs is challenging because of the complex seismic response relationship of these reservoirs. Artificial neural network (ANN) technology has been popular in seismic reservoir prediction because of its self-learning and nonlinear expression abilities. However, problems in the training process of ANNs, such as slow convergence speed and local minima, affect the prediction accuracy. Therefore, this study proposes a hybrid prediction method that combines mutation particle swarm optimization (MPSO) and ANN (MPSO-ANN). It uses the powerful search ability of MPSO to address local optimization problems during training and improve the performance of ANN models in gas-bearing distribution prediction. Furthermore, because the predictions of ANN models require good data sources, multicomponent seismic data that can provide rich gas reservoir information are used as input for MPSO-ANN learning. First, the hyperparameters of the ANN model were analyzed, and ANNs with different structures were constructed. The initial ANN model before optimization exhibited good predictive performance. Then, the parameter settings of MPSO were analyzed, and the MPSO-ANN model was obtained by using MPSO to optimize the weights and biases of the developed ANN model. Finally, the gas-bearing distribution was predicted using multicomponent seismic data. The results indicate that the developed MPSO-ANN model (MSE = 0.0058, RMSE = 0.0762, R2 = 0.9761) has better predictive performance than the PSO-ANN (MSE = 0.0062, RMSE = 0.0786, R2 = 0.9713) and unoptimized ANN models (MSE = 0.0069, RMSE = 0.0833, R2 = 0.9625) on the test dataset. Additionally, the gas-bearing distribution prediction results were consistent overall with the actual drilling results, further verifying the feasibility of this method. The research results may contribute to the application of PSO and ANN in reservoir prediction and other fields.