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Nonlinear differential equations are challenging to solve numerically and areimportant to understanding the dynamics of many physical systems. Deep neuralnetworks have been applied to help alleviate the computational cost that isassociated with solving these systems. We explore the performance and accuracyof various neural architectures on both linear and nonlinear differentialequations 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 setof a linear combination of the Legendre basis, we predict the correspondingcoefficients, $\alpha_i$ which successfully approximate the solution as a sumof smooth basis functions $u \simeq \sum_{i=0}^{N} \alpha_i \varphi_i$. As acomputational example, linear and nonlinear models with Dirichlet or Neumannboundary conditions are considered.

Deep neural network for solving differential equations motivated by Legendre-Galerkin approximation