  Probabilistic solvers for ordinary differential equations (ODEs) have emerged
as an efficient framework for uncertainty quantification and inference on
dynamical systems. In this work, we explain the mathematical assumptions and
detailed implementation schemes behind solving {high-dimensional} ODEs with a
probabilistic numerical algorithm. This has not been possible before due to
matrix-matrix operations in each solver step, but is crucial for scientifically
relevant problems -- most importantly, the solution of discretised {partial}
differential equations. In a nutshell, efficient high-dimensional probabilistic
ODE solutions build either on independence assumptions or on Kronecker
structure in the prior model. We evaluate the resulting efficiency on a range
of problems, including the probabilistic numerical simulation of a differential
equation with millions of dimensions.
