论文标题
深度神经网络，用于解决由Legendre-Galerkin近似动机的微分方程
Deep neural network for solving differential equations motivated by Legendre-Galerkin approximation
论文作者
论文摘要
非线性微分方程在数值上求解具有挑战性，对于理解许多物理系统的动态非常重要。深度神经网络已被应用以帮助减轻与解决这些系统有关的计算成本。我们通过使用光谱元素方法创建准确的训练集，探索线性和非线性微分方程上各种神经体系结构的性能和准确性。接下来，我们实施一种新型的Legendre-Galerkin深神经网络（LGNET）算法，以预测微分方程的解决方案。通过构建一组legendre基础的线性组合，我们预测了相应的系数，即$α_i$，成功将解作为平滑基础函数的总和$ u \ simeq \ sum_ {i = 0}^{n} n}α_iφ_i$。作为计算示例，考虑了具有差异或诺伊曼边界条件的线性和非线性模型。
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is associated with solving these systems. We explore the performance and accuracy of various neural architectures on both linear and nonlinear differential equations by creating accurate training sets with the spectral element method. Next, we implement a novel Legendre-Galerkin Deep Neural Network (LGNet) algorithm to predict solutions to differential equations. By constructing a set of a linear combination of the Legendre basis, we predict the corresponding coefficients, $α_i$ which successfully approximate the solution as a sum of smooth basis functions $u \simeq \sum_{i=0}^{N} α_i φ_i$. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.
