Partial differential equations (PDEs) with multiple scales or those defined over sufficiently large domains arise in various areas of science and engineering and often present problems when approximating the solutions numerically. Machine learning techniques are a relatively recent method for solving PDEs. Despite the increasing number of machine learning strategies developed to approximate PDEs, many remain focused on relatively small domains. When scaling the equations, a large domain is naturally obtained, especially when the solution exhibits multiscale characteristics. This study examines two-scale equations whose solution structures exhibit distinct characteristics: highly localized in some regions and significantly flat in others. These two regions must be adequately addressed over a large domain to approximate the solution more accurately. We focus on the vanishing gradient problem given by the diminishing gradient zone of the activation function over large domains and propose a stratified sampling algorithm to address this problem. We compare the uniform random classical sampling method over the entire domain and the proposed stratified sampling method. The numerical results confirm that the proposed method yields more accurate and consistent solutions than classical methods.

翻译：多尺度偏微分方程或定义在足够大区域上的偏微分方程广泛出现于科学与工程领域，其在数值近似求解时常常带来挑战。机器学习技术是近年来求解偏微分方程的一种相对新颖的方法。尽管为近似求解偏微分方程而开发的机器学习策略日益增多，但许多方法仍局限于相对较小的求解区域。当对方程进行尺度扩展时，尤其是在解呈现多尺度特征的情况下，自然会涉及大区域求解问题。本研究考察解结构呈现显著差异特征的双尺度方程：在某些区域高度局部化，而在其他区域则极为平缓。为了更精确地近似解，必须在大区域上充分处理这两种不同特征的区域。我们关注由大区域上激活函数梯度衰减区所引起的梯度消失问题，并提出一种分层抽样算法以应对该问题。我们比较了全域均匀随机经典抽样方法与所提出的分层抽样方法。数值结果证实，所提方法相比经典方法能够获得更精确且更稳定的解。