In the context of an increasing popularity of data-driven models to represent
dynamical systems, many machine learning-based implementations of the Koopman
operator have recently been proposed. However, the vast majority of those works
are limited to deterministic predictions, while the knowledge of uncertainty is
critical in fields like meteorology and climatology. In this work, we
investigate the training of ensembles of models to produce stochastic outputs.
We show through experiments on real remote sensing image time series that
ensembles of independently trained models are highly overconfident and that
using a training criterion that explicitly encourages the members to produce
predictions with high inter-model variances greatly improves the uncertainty
quantification of the ensembles.

本研究探讨了使用集合模型来生成随机输出的训练方法，证明了通过明确鼓励模型成员产生高模型间方差预测的训练准则，能够显著提高集合模型的不确定性量化能力。

Koopman 集成用于概率时间序列预测

Koopman Ensembles for Probabilistic Time Series Forecasting

Our knowledge of the long-term evolution of solar activity and of its primary
modulation, the 11-year cycle, largely depends on a single direct observational
record: the visual sunspot counts that retrace the last 4 centuries, since the
invention of the astronomical telescope. Currently, this activity index is
available in two main forms: the International Sunspot Number initiated by R.
Wolf in 1849 and the Group Number constructed more recently by Hoyt and
Schatten (1998a,b). Unfortunately, those two series do not match by various
aspects, inducing confusions and contradictions when used in crucial
contemporary studies of the solar dynamo or of the solar forcing on the Earth
climate. Recently, new efforts have been undertaken to diagnose and correct
flaws and biases affecting both sunspot series, in the framework of a series of
dedicated Sunspot Number Workshops. Here, we present a global overview of our
current understanding of the sunspot number calibration. While the early part
of the sunspot record before 1800 is still characterized by large uncertainties
due to poorly observed periods, the more recent sunspot numbers are mainly
affected by three main inhomogeneities: in 1880-1915 for the Group Number and
in 1947 and 1980-2014 for the Sunspot Number. The newly corrected series
clearly indicates a progressive decline of solar activity before the onset of
the Maunder Minimum, while the slowly rising trend of the activity after the
Maunder Minimum is strongly reduced, suggesting that by the mid 18th century,
solar activity had already returned to the level of those observed in recent
solar cycles in the 20th century. We finally conclude with future prospects
opened by this epochal revision of the Sunspot Number, the first one since Wolf
himself, and its reconciliation with the Group Number, a long-awaited
modernization that will feed solar cycle research into the 21st century.

该研究关于太阳活动的长期演化及其主要调制因素，太阳黑子数历史记录的校准问题，以及一些研究关于太阳引起地球气候变化的关键问题。研究表明，太阳活动在小冰期之前逐渐下降，在小冰期之后逐渐回升并稳定。此次研究对太阳黑子数历史记录进行了重大修订，为 21 世纪太阳周期研究提供了宝贵的数据。