Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial and test spaces. Such min-max problem is highly non-linear, and traditional methods often employ different mixed formulations to approximate it. Alternatively, it is possible to address the above saddle-point problem by employing Adversarial Neural Networks: one network approximates the global trial minimum, while another network seeks the test maximizer. However, this approach is numerically unstable due to a lack of continuity of the text maximizers with respect to the trial functions as we approach the exact solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. The resulting Deep Double Ritz Method combines two Neural Networks for approximating the trial and optimal test functions. Numerical results on several 1D diffusion and convection problems support the robustness of our method up to the approximability and trainability capacity of the networks and the optimizer.

基于变压器的语言模型如BERT在大量英语基准上表现出以前的模型，但他们的评估通常限于英语或少量资源的语言。在这项工作中，我们在伯特家族上评估了各种尿潴留的单语，多语言和随机初始化的语言模型，包括爱沙尼亚，芬兰语，匈牙利语，erzya，Moksha，Karelian，Livvi，Komi Permyak，Komi Zyrian，Northern S \' ami，和skolt s''mi。当单晶模型可用时（目前只能等，FI，HU），这些在其母语上表现更好，但一般来说，它们比共享相同字符集的基因无关语言的多语言模型或模型转移。值得注意的是，即使没有特殊努力对封路计优化的特殊努力，高资源模型的直接转移会产生似乎是少数民族尿路语言的艺术POS和NER工具的似乎是有足够的芬特数据的态度。