To approximate the numerical answer of linear second-order regular differential equations with combined boundary situations, we recommend a Deep Learning primarily based totally Galerkin Method primarily based totally on deep neural community getting to know algorithms. Deep getting to know is blended with the Galerkin Method on this method. Instead of mixing linear foundation functions, deep neural networks are used withinside the proposed work. To fulfill the differential operators and boundary situations, we use the gradient descent set of rules to teach the neural community mesh-unfastened with out meshing. Furthermore, the convergence of the loss characteristic and the convergence of the neural community to the precise answer withinside the L2 (Eulidean Distance) norm below sure situations show the approximate cappotential of a neural community. Last however now no longer least, a few numerical experiments display the neural networks' intuitive cappotential to approximate.