Kolmogorov $n$-width

The Kolmogorov $n$-width measures how well some set $\mathcal{M}$ (typically the solution manifold) can be approximated with a linear subspace:

\[d_n(\mathcal{M}) := \mathrm{inf}_{V_n\subset{}V;\mathrm{dim}V_n=n}\mathrm{sup}(u\in\mathcal{M})\mathrm{inf}_{v_n\in{}V_n}|| u - v_n ||_V,\]

with $\mathcal{M}\subset{}V$ and $V$ is a (typically infinite-dimensional) Banach space. For advection-dominated problems (among others) the decay of the Kolmogorov $n$-width is very slow, i.e. one has to pick $n$ very high in order to obtain useful approximations (see [38] and [33]).

In order to overcome this, techniques based on neural networks (see e.g. [39]) and optimal transport (see e.g. [33]) have been used.

References