Reduced-Order Models and Conditional Expectation: Analysing Parametric Low-Order Approximations - Hermann G. Matthies (Computation)
Systems may depend on parameters that can be controlled, serve to optimise the system, are imposed externally, or are uncertain. This last case is taken as the “Leitmotiv” for the following discussion.A reduced-order model is produced from the full-order model through some kind of projection onto a relatively low-dimensional manifold or subspace. The parameter-dependent reduction process produces a function mapping the parameters to 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, a function mapping the parameter set to the image space of the machine learning model is learned from 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 of the conditional expectation—a special case of Bayesian updating, where the Bayesian loss function is the mean square error. This offers the possibility of having a combined view of these methods and also of introducing more general loss functions.