We compare different training strategies for the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight the problems arising from the boundary values. We distinguish between an exact resolution of the boundary values by introducing a distance function and the approximation through a Robin Boundary Value problem. However, distance functions are difficult to obtain for complex domains. Therefore, it is more feasible to solve a Robin Boundary Value problem which approximates the solution to the Dirichlet Boundary Value problem, yet the na\"ive approach to this problem becomes unstable for large penalizations. A novel method to compensate this problem is proposed using a small penalization strength to pre-train the model before the main training on the target penalization strength is conducted. We present numerical and theoretical evidence that the proposed method is beneficial.
翻译:我们比较了深利兹异端方程式与迪里切特边界条件的不同培训策略,并突出了边界价值产生的问题。我们区分了通过引入远程功能准确解决边界价值和通过罗宾边界价值问题近似之间的差别。然而,在复杂的领域很难获得距离功能。因此,更可行的是解决罗宾边界价值问题,这个问题接近于解决迪里切特边界价值问题的办法,然而,对这一问题的现成方法因大规模处罚而变得不稳定。我们建议采用一种新颖的补偿方法,在进行目标惩罚力度主要培训之前,先用少量惩罚力量对模型进行预培训。我们提供了数字和理论证据,证明拟议的方法是有益的。