Quadrature Perturbations and High-Order Time Discretization for Prescribed-Feature Evolutionary GTransNet Galerkin Systems

In this paper we study quadrature perturbations and high-order time discretization for a prescribed-feature evolutionary generalized transferable neural network (Evo-GTransNet) Galerkin method for linear self-adjoint parabolic equations. Assembly quadrature, together with truncated singular value decomposition, is used to select a numerically resolved trial space and obtain coordinates with an identity mass matrix. Since this normalization depends on the assembly quadrature, the retained functions are also evaluated using a separate comparison quadrature, which is finer in the numerical experiments. For the two resulting semidiscrete systems, we derive an exact equation for the difference between their trajectories and an energy estimate that separates errors arising from the effective operator, load vector, and initial projection. The estimate is conditional on the trial space selected by the assembly quadrature. The retained rank, discarded mass, cross mass, and principal angles are used to assess this selection step. Refining the comparison quadrature yields stable retained spaces for the reported configurations, whereas deliberate underintegration leads to much larger perturbations. On the fixed retained systems, the temporal convergence of implicit midpoint, BDF, Radau IIA, and Gauss--Legendre methods is consistent with their classical orders over the resolved refinement ranges. The experiments distinguish the retained-coordinate error from the total solution error; the latter is measured against the PDE solution in the $L^2$ norm, with the error integrals evaluated by sufficiently fine quadrature. They also illustrate the importance of suitable starting values for BDF3 and show how the performance of IMEX-BDF schemes depends on the operator splitting and time step size.