High-Dimensional Uncertainty Propagation

Many uncertainty-propagation methods represent uncertain model inputs as finite-dimensional vectors. Scientific models may instead depend on uncertain coefficient fields, forcing functions, boundary conditions, or geometries. Discretizing these functions can introduce thousands of uncertain variables, making standard finite-dimensional surrogate and sampling methods difficult to use directly.

This chapter develops two complementary responses. We first represent random input and output functions with a small number of coordinates using singular value decomposition, principal component analysis, and the Karhunen–Loève expansion, and then propagate uncertainty through those coordinates. We then learn maps between functions directly with deep operator networks (DeepONets) and, in a companion notebook, Fourier neural operators. Comparing these approaches clarifies when a compact coordinate system is sufficient and when a reusable learned operator is more appropriate. Together, they make uncertainty propagation practical when the scientific model acts on entire fields rather than short parameter vectors.