论文标题
$ r- $自适应深度学习方法解决部分微分方程
$r-$Adaptive Deep Learning Method for Solving Partial Differential Equations
论文作者
论文摘要
我们引入了$ r- $自适应算法，以使用深神经网络求解部分微分方程。所提出的方法限制了张量的产品网格，并在一个维度中优化了边界节点位置，我们从中构建了二维或三维网格。该方法允许定义固定界面设计符合网格，并启用拓扑的更改，即，某些节点可以跳过固定的接口。该方法同时优化了节点位置和PDE解决方案值，而PDE解决方案值在所得网格上。为了在数值上说明我们提出的$ r- $自适应方法的性能，我们将其与搭配方法，最小二乘方法和深层Ritz方法结合使用。我们专注于后者解决一个和二维问题，这些问题的解决方案是平滑，单数和/或表现强的梯度。
We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we build two- or three-dimensional meshes. The method allows the definition of fixed interfaces to design conforming meshes, and enables changes in the topology, i.e., some nodes can jump across fixed interfaces. The method simultaneously optimizes the node locations and the PDE solution values over the resulting mesh. To numerically illustrate the performance of our proposed $r-$adaptive method, we apply it in combination with a collocation method, a Least Squares Method, and a Deep Ritz Method. We focus on the latter to solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients.
