Control of neural transport for normalizing flows

TL;DR

This paper studies how neural network-based velocity fields can control neural transport equations to transform probability densities, providing theoretical guarantees and explicit control constructions with complexity estimates.

Contribution

It proves L^1 approximate controllability for neural transport equations with neural network controls and constructs explicit control vector fields with complexity bounds.

Findings

Any probability density can be approximately transformed into another within any time horizon.

Explicit control vector fields are constructed with quantitative complexity and amplitude estimates.

Provides statistical error bounds using random samples of target densities.

Abstract

Inspired by normalizing flows, we analyze the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfill, at any time instance, a simple neural network ansatz. The L^1 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.