\noindent This work focuses on the efficient resolution of parametric transmission problems in one and two spatial dimensions. These partial differential equations, which model physical systems comprising heterogeneous materials, present significant computational challenges due to the presence of discontinuities across interfaces and singularities at junction points. To address these difficulties, we propose a novel deep learning methodology: the Least-Squares-Based Regularity-Conforming Neural Network (LS-ReCoNN). The proposed method employs a reduced-order adaptive basis based on the Regularity-Conforming Neural Network (ReCoNN) framework \cite{ref_Taylor_Pardo_Munoz}, consisting of two complementary components. The first, a "principal" part, is described by a neural network and captures the smoothly varying aspects of the solution. The second, a ``singular'' component, explicitly models the localized irregularities inherent to junction points using basis functions derived from one-dimensional finite element eigenvalue solvers. To obtain the parameter-dependent solutions, we employ a least-squares optimization strategy adapted from the LS-Net methodology \cite{ref_Baharlouei_Taylor_Uriarte_Pardo}. This approach allows for the efficient approximation of the solution across a broad parameter space by minimizing a quadratic loss function, which serves as a consistent upper bound for the energy-norm error. Numerical experiments demonstrate that the LS-ReCoNN method effectively captures both interfacial discontinuities and junction singularities, significantly reducing the computational expense associated with parametric sweeps while maintaining high accuracy. \begin{center} \fontsize{11}{12}\selectfont \begin{thebibliography}{99} \bibitem{ref_Taylor_Pardo_Munoz} J. M. Taylor, D. Pardo, J. Mu\~noz-Matute, \emph{Regularity-conforming neural networks (ReCoNNs) for solving partial differential equations}, Journal of Computational Physics 532, 113954. \bibitem{ref_Baharlouei_Taylor_Uriarte_Pardo} S. Baharlouei, J. M. Taylor, C. Uriarte, D. Pardo, \emph{A least-squares-based neural network (ls-net) for solving linear parametric pdes}, Computer Methods in Applied Mechanics and Engineering 437, 117757. \end{thebibliography} \end{center}