将参数化降阶模型与条件期望结合，统一分析系统建模与机器学习的内在联系。
Reduced Order Models and Conditional Expectation -- Analysing Parametric Low-Order Approximations
- 通过参数依赖投影构建低维流形上的降阶模型
- 将训练过程视为对条件期望的近似计算
- 为模型优化提供更通用的损失函数框架
系统可能依赖于可调控、优化或外部施加的参数，尤其是不确定性参数被视为核心关注点。通过将全阶模型投影到低维流形或子空间，生成降阶模型；该参数依赖的降阶过程产生一个从参数空间到流形的映射。需分析所有感兴趣参数值下全阶状态与降阶状态的关系。类似地，在机器学习中，模型从训练样本中学习参数集到输出空间的映射，通常最小化均方误差。这些样本可视为来自某个概率分布的采样，因此训练本质上是近似计算期望，得到条件期望的近似，属于贝叶斯更新的一种特例，其中贝叶斯损失函数为均方误差。这为联合分析此类方法提供了可能，并可引入更一般的损失函数。
原文摘要 · Abstract (English)
Systems may depend on parameters which one may control, or which serve to optimise the system, or are imposed externally, or they could be uncertain. This last case is taken as the ``Leitmotiv'' for the following. A reduced order model is produced from the full order model by some kind of projection onto a relatively low-dimensional manifold or subspace. The parameter dependent reduction process produces a function of the parameters into the manifold. One now wants to examine the relation between the full and the reduced state for all possible parameter values of interest. Similarly, in the field of machine learning, also a function of the parameter set into the image space of the machine learning model is learned on a training set of samples, typically minimising the mean-square error. This set may be seen as a sample from some probability distribution, and thus the training is an approximate computation of the expectation, giving an approximation to the conditional expectation, a special case of an Bayesian updating where the Bayesian loss function is the mean-square error. This offers the possibility of having a combined look at these methods, and also of introducing more general loss functions.
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