A globally Lipschitz neural ODE has a global smooth flow

Fix $(t_0,z_0,{\theta }_0)\in I\times {\mathbb{R}}^d\times \mathrm{\Theta }$. Choose a closed time interval and a closed state ball about $(t_0,z_0)$ whose product lies in the open domain; it is compact by [F4]. The continuous function $(t,z)\mapsto \mathrm{\parallel }{D}_{z}v(t,z,{\theta }_0)\mathrm{\parallel }$ is bounded there by [F3]. For each fixed nearby $t$, [F2] on the open state ball gives a state-Lipschitz bound, while smoothness gives continuity of the field. Thus the Picard–Lindelöf hypotheses hold locally at every point of the ODE domain.

For each fixed $\theta$ and initial data, [F5] gives the unique local solution and [F6] gives its unique maximal solution on an open interval. Fix now a nonempty compact $K\subseteq \mathrm{\Theta }$ and a bounded ball $B$ of initial states. For every $\theta \in K$, the assumptions imply $\mathrm{\parallel }v(t,z,\theta ){\mathrm{\parallel }}_2\le M_K+L_K\mathrm{\parallel }z{\mathrm{\parallel }}_2$ on $[0,T]$. For a trajectory starting at $s$ in $B$, its integral equation gives $u(\tau ):=\mathrm{\parallel }z(s+\tau ){\mathrm{\parallel }}_2\le {\sup }_{x\in B}\mathrm{\parallel }x{\mathrm{\parallel }}_2+M_K\tau +L_K{\int }_0^{\tau }u(\sigma )d\sigma$ whenever $s+\tau \in [0,T]$. Apply [F7] with the nondecreasing function $a(\tau )={\sup }_{x\in B}\mathrm{\parallel }x{\mathrm{\parallel }}_2+M_K\tau$; time reversal gives the same bound backward. Thus every such maximal trajectory while it lies in $[0,T]$ has norm at most $R:=({\sup }_{x\in B}\mathrm{\parallel }x{\mathrm{\parallel }}_2+M_{K}T)e^{L_{K}T}$. If a right endpoint $\beta$ of its maximal interval belonged to $(s,T]$, the explicitly defined sequence $t_j=\beta -(\beta -s)\mathrm{/}(j+1)$ would increase to $\beta$, and each graph point $(t_j,z(t_j))$ would lie in the compact cylinder $[0,T]\times {\bar{B}}_R\subset I\times {\mathbb{R}}^d$. By [F8] the solution would extend past $\beta$, contradicting maximality. The explicit sequence $t_j=\alpha +(s-\alpha )\mathrm{/}(j+1)$ gives the same contradiction at any left endpoint $\alpha \in [0,s)$. Therefore every maximal solution contains all of $[0,T]$, and uniqueness follows from [F6]. The sequences are specified by their endpoint and use no choice axiom.

Fix a base initial state, parameter, and times $s,t$; choose compact neighborhoods $B$ and $K$ of the state and parameter. Step 2.1 places the trajectories for these data in one closed bounded state ball, so $[0,T]\times {\bar{B}}_R\times K$ is compact by [F4]. At every point of this cylinder, [F9] and [F10] applied to the smooth augmented field $(z,\theta {)}^{\mathrm{\prime }}=(v(t,z,\theta ),0)$ give an open neighborhood of initial time, state, and parameter, and a positive time radius on which the local transition map is smooth. A finite subcover exists by [F4], and the minimum of its finitely many time radii is positive. Partition the interval from $s$ to $t$ into finitely many segments shorter than this minimum. Each base trajectory node belongs to a selected chart; continuity lets us shrink the initial state and parameter neighborhoods successively so each perturbed node remains in the next chart. The full transition is therefore a finite composition of smooth local maps on a neighborhood of the base data. Since the base data were arbitrary, $(z_s,\theta )\mapsto {\mathrm{\Phi }}_{t,s}^{\theta }(z_s)$ is locally $C^{\mathrm{\infty }}$ for every fixed $s,t$.

By uniqueness from step 2.1, solving from $s$ to $t$ and then from $t$ back to $s$ returns the original initial value; hence ${\mathrm{\Phi }}_{s,t}^{\theta }\circ {\mathrm{\Phi }}_{t,s}^{\theta }$ and ${\mathrm{\Phi }}_{t,s}^{\theta }\circ {\mathrm{\Phi }}_{s,t}^{\theta }$ are both identity maps. Both maps are smooth by step 3.1, so each ${\mathrm{\Phi }}_{t,s}^{\theta }$ is a global $C^{\mathrm{\infty }}$ diffeomorphism with the stated inverse.

Fix $s,\theta ,z_s$. At every interior time $r\in (0,T)$, smoothness of the field and [F10] give a common local interval for nearby initial states at $z_r:={\mathrm{\Phi }}_{r,s}^{\theta }(z_s)$. The hypotheses of [F11] hold there, so $A(t):=D_y{\mathrm{\Phi }}_{t,r}^{\theta }(z_r)$ satisfies $A^{\mathrm{\prime }}(r)=D_{z}v(r,z_r,\theta )$ and $A(r)=I_d$. Uniqueness gives ${\mathrm{\Phi }}_{t,s}^{\theta }={\mathrm{\Phi }}_{t,r}^{\theta }\circ {\mathrm{\Phi }}_{r,s}^{\theta }$ near $t=r$; differentiating this identity in $z_s$ using [F12] yields $J_{t,s}=A(t)J_{r,s}$. Differentiating at $t=r$ therefore gives $\frac{d}{dt}{J}_{t,s}=D_{z}v(r,{\mathrm{\Phi }}_{r,s}^{\theta }(z_s),\theta )J_{r,s}$. Since $r$ was arbitrary this holds on $(0,T)$; the local transitions at $0,T$ give the corresponding one-sided derivatives. At $t=s$, the transition is the identity and so $J_{s,s}=I_d$.