论文标题
使用神经网络解决部分微分方程的深度双Ritz方法（D $^2 $ RM）
A Deep Double Ritz Method (D$^2$RM) for solving Partial Differential Equations using Neural Networks
论文作者
论文摘要
残留最小化是一种广泛使用的技术，用于以各种形式求解部分微分方程。它最大程度地减少了残差的双重规范，该规范自然会在所谓的试验和测试空间中产生鞍点（Min-Max）问题。在神经网络的背景下，我们可以通过使用一个网络来寻求最低试验，而另一个网络则寻求测试最大值，我们可以通过使用一个网络来解决这种最小的方法。但是，当我们接近试验解决方案时，所得的方法在数值上是不稳定的。为了克服这一点，我们将剩余的最小化重新制定为从另一个RITZ功能最小化计算出的最佳测试功能的丽思疗法的等效最小化。我们将结果方案称为Deep Double Ritz方法（D $^2 $ RM），该方法结合了两个神经网络，用于近似试验功能和沿嵌套的双Ritz最小化策略的最佳测试功能。关于不同扩散和对流问题的数值结果支持我们方法的鲁棒性，直至网络的近似特性以及优化器的训练能力。
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. In the context of neural networks, we can address this min-max approach by employing one network to seek the trial minimum, while another network seeks the test maximizers. However, the resulting method is numerically unstable as we approach the trial 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. We call the resulting scheme the Deep Double Ritz Method (D$^2$RM), which combines two neural networks for approximating trial functions and optimal test functions along a nested double Ritz minimization strategy. Numerical results on different diffusion and convection problems support the robustness of our method, up to the approximation properties of the networks and the training capacity of the optimizers.
