Accurate uncertainty quantification is necessary to enhance the reliability of deep learning models in real-world applications. In the case of regression tasks, prediction intervals (PIs) should be provided along with the deterministic predictions of deep learning models. Such PIs are useful or "high-quality" as long as they are sufficiently narrow and capture most of the probability density. In this paper, we present a method to learn prediction intervals for regression-based neural networks automatically in addition to the conventional target predictions. In particular, we train two companion neural networks: one that uses one output, the target estimate, and another that uses two outputs, the upper and lower bounds of the corresponding PI. Our main contribution is the design of a novel loss function for the PI-generation network that takes into account the output of the target-estimation network and has two optimization objectives: minimizing the mean prediction interval width and ensuring the PI integrity using constraints that maximize the prediction interval probability coverage implicitly. Furthermore, we introduce a self-adaptive coefficient that balances both objectives within the loss function, which alleviates the task of fine-tuning. Experiments using a synthetic dataset, eight benchmark datasets, and a real-world crop yield prediction dataset showed that our method was able to maintain a nominal probability coverage and produce significantly narrower PIs without detriment to its target estimation accuracy when compared to those PIs generated by three state-of-the-art neural-network-based methods. In other words, our method was shown to produce higher-quality PIs.

翻译：准确的量化不确定性对于提升深度学习模型在实际应用中的可靠性至关重要。在回归任务中，预测区间应随深度学习模型的确定性预测一同提供。此类预测区间应足够窄且能捕获大部分概率密度，才被视为有用或“高质量”。本文提出一种方法，使基于回归的神经网络能在常规目标预测之外自动学习预测区间。具体而言，我们训练两个配套神经网络：一个输出目标估计值，另一个输出对应预测区间的上下界。主要贡献在于设计了一种新颖的损失函数用于预测区间生成网络，该函数结合了目标估计网络的输出，并包含两个优化目标：最小化平均预测区间宽度，并通过约束条件隐式最大化预测区间概率覆盖度以确保其完整性。此外，我们引入一个自适应系数来平衡损失函数中的两个目标，从而简化了微调过程。在合成数据集、八个基准数据集以及一个真实作物产量预测数据集上的实验表明，与三种基于神经网络的最新方法生成的预测区间相比，我们的方法能够维持名义概率覆盖度，同时生成显著更窄的预测区间，且不损害目标估计精度。换言之，该方法生成了更高质量的预测区间。