论文标题
ODE解决方案类别：平滑度，覆盖数，对嘈杂功能拟合的影响以及平滑度现象的诅咒
Classes of ODE solutions: smoothness, covering numbers, implications for noisy function fitting, and the curse of smoothness phenomenon
论文作者
论文摘要
从数据中恢复ODE解决方案的许多数值方法都取决于使用基本函数或在最小平方标准下使用内核函数近似解决方案。这种方法的准确性取决于解决方案的平稳性。本文通过建立有关ode解决方案类别的平滑度和涵盖数量（以衡量其“大小”的量度）来为这些方法提供理论基础。我们的结果为“平滑度和一类ODES的“大小”如何影响相关解决方案类别的“大小”）提供了答案？我们表明：（1）对于$ y^{'} = f \ left（y \ right）$和$ y^{'} = f \ left（x，x，x，\，y \ right）$，如果所有$ k $ th（$ k \ leqleqβ+1 $ 1 $ 1 $）订单的订单衍生物的$ 1 $ the $ the $ th in $ th in $ th（k）int $ th（k），然后$ th（k），然后k. $ k $中的进度快速 - “光滑的诅咒”； （2）我们的上限$（β+2） - $度平滑溶液类别大于“标准” $（β+2）的上限 - 单变量函数的$度平滑类别； （3）噪声恢复的最小二乘拟合的平均平方误差的收敛速率不超过$ \左（\ frac {1} {n} {n} {n} \ right）^{\ frac {2 \ left（β+2 \ right）} {2 $ n =ω\ left（\ left（β\ sqrt {\ log \ left（β\ vee1 \ right）} \ right）^{4β+10} \ right）$，在这种情况下，速率$ \ left（\ frac {1} {n} \ right）^{\ frac {\ frac {2 \ left（β+2 \ right）} {2 \ left（β+2 \ right）+1}} $在$ y^{'} = f \ weft（'} = f \ weft（x，x，x，y，y，y）的情况下， （4）更一般地，对于高阶Picard类型ODE，$ y^{\ left（m \右）} = f \ left（x，x，x，y，y，\，\，y^{'}，\，\，...，y^{\ lest（m-1 \ right）} $ \ MATHCAL {f} $该$ f $范围超过了，并且初始值所在的集合的覆盖号。
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 $y^{'}=f\left(y\right)$ and $y^{'}=f\left(x,\,y\right)$, if the absolute values of all $k$th ($k\leqβ+1$) 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 $\left(\frac{1}{n}\right)^{\frac{2\left(β+2\right)}{2\left(β+2\right)+1}}$ if $n=Ω\left(\left(β\sqrt{\log\left(β\vee1\right)}\right)^{4β+10}\right)$, and under this condition, the rate $\left(\frac{1}{n}\right)^{\frac{2\left(β+2\right)}{2\left(β+2\right)+1}}$ is minimax optimal in the case of $y^{'}=f\left(x,\,y\right)$; (4) more generally, 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 product of the covering number of the class $\mathcal{F}$ that $f$ ranges over and the covering number of the set where initial values lie.
