Uncertainty quantification in numerical weather and climate prediction is usually achieved using a Monte Carlo estimation (i.e., ensemble forecasting) of the forecast probability distribution function of the state of the system. In this work, we present a method for uncertainty quantification based on neural networks and using a likelihood-based loss function to train the network. This provides state dependent uncertainty estimation, without the need of integrating an ensemble of forecasts. The method is evaluated with a chaotic low-dimensional model in two scenarios: with stochastic errors only (SE) and systematic and stochastic errors (SSE).