Reduced Order Models and Conditional Expectation -- Analysing Parametric Low-Order Approximations

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.