Deep neural network machine learning models have demonstrated success in addressing time-dependent Partial Differential Equations (PDEs, e.g., Physics Informed Neural Networks or PINNs and mesh graph network models by Google DeepMind). Yet, these network models encounter two significant challenges: (1) generalisation to problems beyond their training data and (2) numerical stability during long-time evolutions. In this talk a new spectral framework based on a local error analysis is presented to design and optimise numerical methods for convection problems called Local Transfer Analysis (LTA). LTA converts traditional numerical discretisation to a network of impedance blocks where parameters can be introduced to tune the local block impedance. Such a network's impedance can be tuned using a Deep Graph Network that predicts optimal values for the parameters that lead to matched impedance. This allows locally tuned traditional numerical schemes that do not suffer from stability problems, at the same time generalises to a wide range of problems outside of training on unstructured meshes outperforming the unoptimised scheme. Application of the framework to tune and optimise the Two-step Taylor Galerkin scheme (TTGC) used extensively in CERFACS for Combustion LES problems is presented for linear convection, inviscid Burgers' and Euler equations on unstructured meshes.