论文标题
深彼得 - 加利尔金方法用于求解部分微分方程
Deep Petrov-Galerkin Method for Solving Partial Differential Equations
论文作者
论文摘要
深度神经网络是近似功能的强大工具，它们可用于成功解决许多领域的各种问题。在本文中，我们提出了一种基于神经网络的数值方法来求解部分微分方程。在这个新框架中，该方法是根据弱公式设计的，未知功能由深神网络近似，可以通过不同的方法选择测试功能，例如，有限元方法的基础函数，神经网络等。由于试验功能和测试功能的空间不同，因此我们通过深彼得罗夫 - 加利尔金方法（DPGM）命名了这种新方法。所得的线性系统不一定是对称和正方形的，因此通过最小二乘方法解决了离散的问题。以泊松问题为例，也提出并研究了基于几种混合配方的混合DPGM。此外，我们将DPGM应用于基于时空方法的两个经典时间依赖性问题，也就是说，未知函数由神经网络近似，在该函数中，在该函数中，时间变量和空间变量被同样治疗，并且初始条件被视为时空域的边界条件。最后，提出了几个数值示例以显示DPGM的性能，我们观察到这种新方法在几个方面都优于传统的数值方法。
Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential equations. In this new framework, the method is designed on weak formulations, and the unknown functions are approximated by deep neural networks and test functions can be chosen by different approaches, for instance, basis functions of finite element methods, neural networks, and so on. Because the spaces of trial function and test function are different, we name this new approach by Deep Petrov-Galerkin Method (DPGM). The resulted linear system is not necessarily to be symmetric and square, so the discretized problem is solved by a least-square method. Take the Poisson problem as an example, mixed DPGMs based on several mixed formulations are proposed and studied as well. In addition, we apply the DPGM to solve two classical time-dependent problems based on the space-time approach, that is, the unknown function is approximated by a neural network, in which temporal variable and spatial variables are treated equally, and the initial conditions are regarded as boundary conditions for the space-time domain. Finally, several numerical examples are presented to show the performance of the DPGMs, and we observe that this new method outperforms traditional numerical methods in several aspects.
