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Many numerical methods for recovering ODE solutions from data rely onapproximating the solutions using basis functions or kernel functions under aleast square criterion. The accuracy of this approach hinges on the smoothnessof the solutions. This paper provides a theoretical foundation for thesemethods by establishing novel results on the smoothness and covering numbers ofODE solution classes (as a measure of their "size"). Our results provideanswers to "how do the degree of smoothness and the "size" of a class of ODEsaffect the "size" of the associated class of solutions?" We show that: (1) for$y^{'}=f\left(y\right)$ and $y^{'}=f\left(x,\,y\right)$, if the absolute valuesof all $k$th ($k\leq\beta+1$) order derivatives of $f$ are bounded by $1$, thenthe solution can end up with the $(k+1)$th derivative whose magnitude growsfactorially fast in $k$ -- "a curse of smoothness"; (2) our upper bounds forthe covering numbers of the $(\beta+2)-$degree smooth solution classes aregreater than those of the "standard" $(\beta+2)-$degree smooth class ofunivariate functions; (3) the mean squared error of least squares fitting fornoisy recovery has a convergence rate no larger than$\left(\frac{1}{n}\right)^{\frac{2\left(\beta+2\right)}{2\left(\beta+2\right)+1}}$if$n=\Omega\left(\left(\beta\sqrt{\log\left(\beta\vee1\right)}\right)^{4\beta+10}\right)$,and under this condition, the rate$\left(\frac{1}{n}\right)^{\frac{2\left(\beta+2\right)}{2\left(\beta+2\right)+1}}$is minimax optimal in the case of $y^{'}=f\left(x,\,y\right)$; (4) moregenerally, for the higher order Picard type ODEs,$y^{\left(m\right)}=f\left(x,\,y,\,y^{'},\,...,y^{\left(m-1\right)}\right)$,the covering number of the solution class is bounded from above by the productof the covering number of the class $\mathcal{F}$ that $f$ ranges over and thecovering number of the set where initial values lie.

Classes of ODE solutions: smoothness, covering numbers, implications for noisy function fitting, and the curse of smoothness phenomenon