  Ordinary differential equations (ODEs), via their induced flow maps, provide
a powerful framework to parameterize invertible transformations for the purpose
of representing complex probability distributions. While such models have
achieved enormous success in machine learning, particularly for generative
modeling and density estimation, little is known about their statistical
properties. This work establishes the first general nonparametric statistical
convergence analysis for distribution learning via ODE models trained through
likelihood maximization. We first prove a convergence theorem applicable to
arbitrary velocity field classes $\mathcal{F}$ satisfying certain simple
boundary constraints. This general result captures the trade-off between
approximation error (`bias') and the complexity of the ODE model (`variance').
We show that the latter can be quantified via the $C^1$-metric entropy of the
class $\mathcal F$. We then apply this general framework to the setting of
$C^k$-smooth target densities, and establish nearly minimax-optimal convergence
rates for two relevant velocity field classes $\mathcal F$: $C^k$ functions and
neural networks. The latter is the practically important case of neural ODEs.
  Our proof techniques require a careful synthesis of (i) analytical stability
results for ODEs, (ii) classical theory for sieved M-estimators, and (iii)
recent results on approximation rates and metric entropies of neural network
classes. The results also provide theoretical insight on how the choice of
velocity field class, and the dependence of this choice on sample size $n$
(e.g., the scaling of width, depth, and sparsity of neural network classes),
impacts statistical performance.
