  In this paper, we perform the convergence analysis of unsupervised
Legendre--Galerkin neural networks (ULGNet), a deep-learning-based numerical
method for solving partial differential equations (PDEs). Unlike existing deep
learning-based numerical methods for PDEs, the ULGNet expresses the solution as
a spectral expansion with respect to the Legendre basis and predicts the
coefficients with deep neural networks by solving a variational residual
minimization problem. Since the corresponding loss function is equivalent to
the residual induced by the linear algebraic system depending on the choice of
basis functions, we prove that the minimizer of the discrete loss function
converges to the weak solution of the PDEs. Numerical evidence will also be
provided to support the theoretical result. Key technical tools include the
variant of the universal approximation theorem for bounded neural networks, the
analysis of the stiffness and mass matrices, and the uniform law of large
numbers in terms of the Rademacher complexity.
