Exact empirical-measure transport controllability for linear-control neural ODEs with ReLU activation

Establish whether, for the neural ODE with linear-in-parameter vector field v(x,θ)=wσ(x)+b where σ(x)=(x)_+ (ReLU) and θ=(w,b)∈R^{d×d}×R^d, one can exactly transport any empirical measure μ0=(1/n)∑_{i=1}^n δ_{x^{(i)}} to any target empirical measure μ1=(1/n)∑_{i=1}^n δ_{y^{(i)}} by choosing time-dependent parameters θ(t) on [0,T], i.e., determine the existence of θ(t) such that the flow map Φ^T_θ satisfies Φ^T_θ(x^{(i)})=y^{(i)} for all i∈[n].

Background

The paper studies controllability of the continuity equation arising in normalizing flows and, in a separate discussion, considers a neural ODE with a vector field linear in the parameters, v(x,θ)=wσ(x)+b, where σ is applied component-wise. For smooth activation σ, geometric control techniques (via iterated Lie brackets and Chow–Rashevskii) can achieve exact steering of finitely many points, hence transporting an empirical measure to another.

However, for the non-smooth ReLU activation σ(x)=(x)_+, the applicability of such controllability results is explicitly stated as not known at the point of discussion. This raises the question of whether exact point-to-point steering—and thus exact empirical-measure transport—is possible with the linear-control ReLU model.