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This paper introduces a framework for capturing stochasticity of choice probabilities in neural networks, derived from and fully consistent with the Random Utility Maximization (RUM) theory, referred to as RUM-NN. Neural network models show remarkable performance compared with statistical models; however, they are often criticized for their lack of transparency and interoperability. The proposed RUM-NN is introduced in both linear and nonlinear structures. The linear RUM-NN retains the interpretability and identifiability of traditional econometric discrete choice models while using neural network-based estimation techniques. The nonlinear RUM-NN extends the model's flexibility and predictive capabilities to capture nonlinear relationships between variables within utility functions. Additionally, the RUM-NN allows for the implementation of various parametric distributions for unobserved error components in the utility function and captures correlations among error terms. The performance of RUM-NN in parameter recovery and prediction accuracy is rigorously evaluated using synthetic datasets through Monte Carlo experiments. Additionally, RUM-NN is evaluated on the Swissmetro and the London Passenger Mode Choice (LPMC) datasets with different sets of distribution assumptions for the error component. The results demonstrate that RUM-NN under a linear utility structure and IID Gumbel error terms can replicate the performance of the Multinomial Logit (MNL) model, but relaxing those constraints leads to superior performance for both Swissmetro and LPMC datasets. By introducing a novel estimation approach aligned with statistical theories, this study empowers econometricians to harness the advantages of neural network models.