Deep learning models are often trained to approximate dynamical systems that can be modeled using differential equations. Many of these models are optimized to predict one step ahead; such approaches produce calibrated one-step predictions if the predictive model can quantify uncertainty, such as Deep Ensembles. At inference time, multi-step predictions are generated via autoregression, which needs a sound uncertainty propagation method to produce calibrated multi-step predictions. This work introduces an alternative Predictor-Corrector approach named \hop{} that uses Modern Hopfield Networks (MHN) to learn the errors of a deterministic Predictor that approximates the dynamical system. The Corrector predicts a set of errors for the Predictor's output based on a context state at any timestep during autoregression. The set of errors creates sharper and well-calibrated prediction intervals with higher predictive accuracy compared to baselines without uncertainty propagation. The calibration and prediction performances are evaluated across a set of dynamical systems. This work is also the first to benchmark existing uncertainty propagation methods based on calibration errors.

翻译：深度学习模型常被训练来逼近可用微分方程建模的动力学系统。许多此类模型被优化为单步预测；若预测模型能够量化不确定性（如深度集成方法），这类方法可产生校准的单步预测。在推理阶段，多步预测通过自回归生成，这需要可靠的不确定性传播方法以产生校准的多步预测。本文提出一种名为\hop{}的预测器-校正器替代方法，该方法利用现代Hopfield网络（MHN）学习逼近动力学系统的确定性预测器所产生的误差。校正器基于自回归过程中任意时间步的上下文状态，预测一组针对预测器输出的误差。与未进行不确定性传播的基线方法相比，该误差集合能生成更尖锐、校准更优的预测区间，并具有更高的预测精度。本文在一系列动力学系统上评估了校准与预测性能。此项工作也是首个基于校准误差对现有不确定性传播方法进行系统性基准测试的研究。