论文标题
高维部分微分方程的时空深神经网络近似
Space-time deep neural network approximations for high-dimensional partial differential equations
论文作者
论文摘要
它是应用数学中最具挑战性的问题之一，可以求解高维偏微分方程（PDE），并且在科学文献中，PDE的PDE的大多数数值近似方法都遭受了所谓的尺寸的诅咒，即在相应的近似值中使用相应的计算操作的数量，以获得相应的近似值，以获得相应的近似值。在$ \ varepsilon $的PDE尺寸和/或倒数中。最近，已经提出了某些基于深度学习的PDE的基于深度学习的近似方法，并且对这种方法的各种数值模拟表明，深度神经网络（DNN）近似值可能确实能够克服维数的诅咒，从而，在两个近似值中都在pde $ dnns中生长的真实参数的数量，在pde $ ddnns中生长$ d;处方精度的倒数$ \ varepsilon> 0 $。现在，科学文献中也有一些严格的结果，通过证明DNN在近似PDE的溶液中克服了维度的诅咒，从而证实了这种猜想。这些结果中的每一个都表明，DNN在固定时间点$ t> 0 $和紧凑型立方体$ [a，b]^d $上近似合适的PDE解决方案时都克服了维数的诅咒，但这些结果都没有提供任何问题的答案，该问题是否可以解决整个PDE解决方案在$ [a，b] \ times [a，b] \ timesimentive by unimistion cul ynofe by unimistion cul y nortimistion cube [a，b]^d $ undimistion culimistion culimistion culimistion。克服这个问题正是本文的主题。更具体地说，这项工作的主要结果尤其证明了\ mathbb {r} $，$ b \ in（a，\ infty）$中的每一个$ a \ in（a，\ infty）$的证明，dnns可以近似于某些Kolmogorov PDES的解决方案在时空区域$ [a，b]^a，b]^d $ cursemental cursemental cursemental cursemental y dimemental y dimession y dimemental y dimemental y dimemental y dimemental y dimemental y dimemental y dimesional y dimemental of dnns的解决方案。
It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision $\varepsilon>0$ grows exponentially in the PDE dimension and/or the reciprocal of $\varepsilon$. Recently, certain deep learning based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension $d\in\mathbb{N}$ and the reciprocal of the prescribed accuracy $\varepsilon>0$. There are now also a few rigorous results in the scientific literature which substantiate this conjecture by proving that DNNs overcome the curse of dimensionality in approximating solutions of PDEs. Each of these results establishes that DNNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point $T>0$ and on a compact cube $[a,b]^d$ in space but none of these results provides an answer to the question whether the entire PDE solution on $[0,T]\times [a,b]^d$ can be approximated by DNNs without the curse of dimensionality. It is precisely the subject of this article to overcome this issue. More specifically, the main result of this work in particular proves for every $a\in\mathbb{R}$, $ b\in (a,\infty)$ that solutions of certain Kolmogorov PDEs can be approximated by DNNs on the space-time region $[0,T]\times [a,b]^d$ without the curse of dimensionality.
