Autonomous-Flow-Based Generation

arXiv:2511.09902v3 Announce Type: replace Abstract: We show that using autonomous-flow-based generation, one can universally approximate orientation-preserving diffeomorphisms defined on the cube by Neural ODEs with rate 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_id condition on (0,1)^d, we show that using autonomous-flow-based generation, one can universally approximate compactly supported_id diffeomorphisms on (0,1)^d for any dimension with rate O((frac{P}{log P})^{-2/d}) with P parameters and for compactly supported_id homeomorphisms on (0,1)^d in dimension d geq 5 with rate O(P^{-1/(d+1)}) with P parameters and by a composition of at most I_d autonomous Neural ODEs with the same support_id, where I_d depends only on the dimension. Moreover, we show that the class of single autonomous flows compactly supported_id on (0,1)^d is meagre in the space of compactly supported_id homeomorphisms on (0,1)^d for dge 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 O(P^{-1/(d+1)}) with P parameters.

Source: arXiv cs.LG | 2026-09-11