  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, $\alpha_i$ which successfully approximate the solution as a sum
of smooth basis functions $u \simeq \sum_{i=0}^{N} \alpha_i \varphi_i$. As a
computational example, linear and nonlinear models with Dirichlet or Neumann
boundary conditions are considered.
