✨ TL;DR
This paper presents a neural network framework that simultaneously generates point forecasts and prediction intervals for multi-step time series forecasting, using multi-objective optimization to automatically balance forecast accuracy and interval sharpness while guaranteeing non-crossing intervals and target coverage. The method eliminates manual hyperparameter tuning and demonstrates superior performance on solar irradiance forecasting compared to existing approaches.
Probabilistic forecasting requires generating both point forecasts and prediction intervals (PIs) that satisfy coverage guarantees while being as narrow as possible. Existing methods face several challenges: they often produce crossing prediction intervals that violate logical consistency, rely on manual tuning of weights in scalarized loss functions to balance multiple objectives (point accuracy, coverage probability, and interval sharpness), and may fail to guarantee target coverage probability. The trial-and-error process of hyperparameter tuning is time-consuming and prevents these methods from being universally applicable across different forecasting problems and scales. Additionally, current approaches typically treat point and interval forecasting as separate tasks or use ad-hoc methods to combine them, lacking a principled framework that can simultaneously optimize all objectives while respecting hard constraints like non-crossing intervals and coverage guarantees. This makes it difficult to deploy robust probabilistic forecasting systems in practice.
The paper formulates point and interval forecasting as a multi-objective optimization problem solved through multi-gradient descent, which adaptively determines optimal weights for different loss components without manual tuning. The core innovation is a new prediction interval loss function based on an extended log-barrier method with an adaptive hyperparameter that strictly enforces target coverage probability (PICP) while maximizing sharpness. This loss function is scale-independent, making it universally applicable across different datasets. The architecture uses a hybrid design with a shared temporal model (applicable to any deep learning backbone like LSTM or Transformer) combined with horizon-specific submodels for different forecast horizons. The model structure inherently ensures non-crossing prediction intervals through its design. The training strategy leverages multi-gradient descent to automatically balance three objectives: point forecast accuracy, interval coverage, and interval sharpness, eliminating the need for manual weight selection in the loss function.