Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients. The localized orthogonal decomposition (LOD) method is an effective way of dealing with such coefficients, especially if they are non-periodic and non-smooth. It modifies classical finite element basis functions by suitable fine-scale corrections. In this paper, we make use of the structure of the LOD method, but we propose to calculate the corrections based on a Deep Ritz approach involving a parametrization of the coefficients to tackle temporal variations or uncertainties. Numerical examples for a parabolic model problem are presented to assess the performance of the approach.

翻译：数值均匀化方法旨在为涉及高度振荡系数的（椭圆型）偏微分方程提供适当的粗尺度近似解。局部正交分解（LOD）方法是处理此类系数的有效手段，尤其适用于非周期、非光滑系数情形。该方法通过引入适当的细尺度修正来改进经典有限元基函数。本文利用LOD方法的结构框架，提出基于深度Ritz方法计算修正项，通过对系数进行参数化处理以应对时变或不确定性问题。文中通过抛物型模型问题的数值算例验证了该方法的性能表现。