We show that using autonomous-flow-based generation, one can universally approximate orientation-preserving diffeomorphisms defined on the cube by Neural ODEs with rate $\mathcal{O}(P^{-1/d})$ with $P$ parameters. On the other hand, we show that by using only a single autonomous flow, the class of Neural ODEs is nowhere dense on the cube in dimension $d \ge 2$ . Under a compact-support$_\mathrm{id}$ condition on $(0,1)^d$, we show that using autonomous-flow-based generation, one can universally approximate compactly supported$_\mathrm{id}$ diffeomorphisms on $(0,1)^d$ for any dimension with rate $\mathcal{O}((\frac{P}{\log P})^{-2/d})$ with $P$ parameters and for compactly supported$_\mathrm{id}$ homeomorphisms on $(0,1)^d$ in dimension $d \geq 5$ with rate $\mathcal{O}(P^{-1/(d+1)})$ with $P$ parameters and by a composition of at most $I_d$ autonomous Neural ODEs with the same support$_\mathrm{id}$, where $I_d$ depends only on the dimension. Moreover, we show that the class of single autonomous flows compactly supported$_\mathrm{id}$ on $(0,1)^d$ is meagre in the space of compactly supported$_\mathrm{id}$ homeomorphisms on $(0,1)^d$ for $d\ge 2$. By linearly lifting the domain into one higher dimension, we obtain a universal approximation result for Lipschitz functions compactly supported on $(0,1)^d$ with rate $\mathcal{O}(P^{-1/(d+1)})$ with $P$ parameters.

翻译：暂无翻译