Instantaneous change of variables along a neural ODE flow

Assume Countable Choice. Let $T>0$, let $d,q\ge 1$, let $\mathrm{\Theta }\subseteq {\mathbb{R}}^q$ be open, let $I\subseteq \mathbb{R}$ be an open interval containing $[0,T]$, and let $v:I\times {\mathbb{R}}^d\times \mathrm{\Theta }\to {\mathbb{R}}^d$ be jointly $C^{\mathrm{\infty }}$. For every compact $K\subseteq \mathrm{\Theta }$, assume there are constants $L_K,M_K\ge 0$ such that, for all $t\in [0,T]$, $\eta \in K$, and $z,w\in {\mathbb{R}}^d$,

$\mathrm{\parallel }v(t,z,\eta )-v(t,w,\eta ){\mathrm{\parallel }}_2\le L_K\mathrm{\parallel }z-w{\mathrm{\parallel }}_2,\mathrm{\parallel }v(t,0,\eta ){\mathrm{\parallel }}_2\le M_K\mathrm{.}$

Fix $\theta \in \mathrm{\Theta }$ and let ${\mathrm{\Phi }}_t:={\mathrm{\Phi }}_{t,0}^{\theta }:{\mathbb{R}}^d\to {\mathbb{R}}^d$ be its flow. Let $p_0:{\mathbb{R}}^d\to (0,\mathrm{\infty })$ be a continuous probability density with respect to Lebesgue measure, define ${\mu }_0(E):={\int }_E{p}_0(z)dz$ for Borel $E$, and define the pushforward measure ${\mu }_t(E):={\mu }_0({\mathrm{\Phi }}_t^{-1}(E))$. For $y\in {\mathbb{R}}^d$, set

$p_t(y):=\frac{p_0({\mathrm{\Phi }}_t^{-1}(y))}{\mathrm{\mid }\det D{\mathrm{\Phi }}_t({\mathrm{\Phi }}_t^{-1}(y))\mathrm{\mid }}\mathrm{.}$

Then $p_t$ is a density of ${\mu }_t$. For every $z_0\in {\mathbb{R}}^d$, write $z_t:={\mathrm{\Phi }}_t(z_0)$ and $J_t:=D{\mathrm{\Phi }}_t(z_0)$; the determinant of $J_t$ is positive and

$p_t({\mathrm{\Phi }}_t(z_0))=\frac{p_0(z_0)}{\det J_t}>0,\frac{d}{dt}\log p_t(z_t)=-\mathrm{tr}(D_{z}v(t,z_t,\theta ))=-{\sum }_{i=1}^d{\mathrm{\partial }}_{z_i}{v}_i(t,z_t,\theta ),$

The derivative identity holds for $0<t<T$ and has one-sided meanings at $0$ and $T$.

Chen et al., §4 and Appendix A

Papamakarios et al., §4.1