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

arXiv: https://arxiv.org/abs/2011.11371 · HTML: https://mlsanity.com/paper/2011.11371/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2020.

## Built on by (most cited first, 1 of 1)

- [Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs](https://mlsanity.com/paper/2606.27285/) (2026)

## Abstract (arXiv)

Many numerical methods for recovering ODE solutions from data rely on approximating the solutions using basis functions or kernel functions under a least square criterion. The accuracy of this approach hinges on the smoothness of the solutions. This paper provides a theoretical foundation for these methods by establishing novel results on the smoothness and covering numbers of ODE solution classes (as a measure of their "size"). Our results provide answers to "how do the degree of smoothness and the "size" of a class of ODEs affect the "size" of the associated class of solutions?" We show that: (1) for y'=fleft(yright) and y'=fleft(x,,yright), if the absolute values of all kth (k≤β+1) order derivatives of f are bounded by 1, then the solution can end up with the (k+1)th derivative whose magnitude grows factorially fast in k -- "a curse of smoothness"; (2) our upper bounds for the covering numbers of the (β+2)-degree smooth solution classes are greater than those of the "standard" (β+2)-degree smooth class of univariate functions; (3) the mean squared error of least squares fitting for noisy recovery has a convergence rate no larger than left(frac1nright)^frac2left(β+2right)2left(β+2right)+1 if n=Omegaleft(left(β√logleft(βvee1right)right)4β+10right), and under this condition, the rate left(frac1nright)^frac2left(β+2right)2left(β+2right)+1 is minimax optimal in the case of y'=fleft(x,,yright); (4) more generally, for the higher order Picard type ODEs, yleft(mright)=fleft(x,,y,,y',,...,yleft(m-1right)right), the covering number of the solution class is bounded from above by the product of the covering number of the class F that f ranges over and the covering number of the set where initial values lie.