Control of neural transport for normalising flows

Inspired by normalising flows, we analyse the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfil, at any time instance, a simple neural network ansatz. The L1 approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available. (c) 2023 Elsevier Masson SAS. All rights reserved.

Automated summary

Inspired by normalising flows, we analyze the bilinear control of neural transport equations using time-dependent velocity fields constrained by a simple neural network assumption. We prove the L1 approximate controllability property, showing that any probability density can be driven arbitrarily close to any other one within any given time horizon. The control vector fields are explicitly and recursively constructed, providing quantitative estimates of their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.