In this paper, we study the error in first order Sobolev norm in the approximation of solutions to linear parabolic PDEs. We use a Monte Carlo Euler scheme obtained from combining the Feynman--Kac representation with a Euler discretization of the underlying stochastic process. We derive approximation rates depending on the time-discretization, the number of Monte Carlo simulations, and the dimension. In particular, we show that the Monte Carlo Euler scheme breaks the curse of dimensionality with respect to the first order Sobolev norm. Our argument is based on new estimates on the weak error of the Euler approximation of a diffusion process together with its derivative with respect to the initial condition. As a consequence, we obtain that neural networks are able to approximate solutions of linear parabolic PDEs in first order Sobolev norm without the curse of dimensionality if the coefficients of the PDEs admit an efficient approximation with neural networks.

翻译：本文研究线性抛物型偏微分方程解近似中的一阶Sobolev范数误差。我们采用基于Feynman-Kac表示与底层随机过程欧拉离散化相结合的蒙特卡洛欧拉格式，推导出依赖时间离散化、蒙特卡洛模拟次数和维度的近似速率。特别地，我们证明该蒙特卡洛欧拉格式在一阶Sobolev范数意义下突破了维度灾难。论证基于扩散过程欧拉逼近及其关于初始条件的导数的弱误差新估计。由此可得：若偏微分方程系数允许神经网络高效近似，则神经网络能够在一阶Sobolev范数下无维度灾难地逼近线性抛物型偏微分方程的解。